What single, pure compound would best replace petrol?
Higher $\mathrm{RON}$ seems possible. Yet the boiling point rises accordingly while heat of combustion remains roughly the same. Here are two compounds that fit all posed criteria.

\begin{array}{|c|c|c|c|c|}
\hline
\mathbf{Molecule} &amp; \mathrm{mp\ \mathrm{(^\circ C)}}  &amp; \mathrm{bp\ \mathrm{(^\circ C)}} &amp; \Delta_\mathrm cH_\mathrm m^\circ\ (\mathrm{kJ/mol}) \ \text{@}25\ ^\circ \mathrm{C} &amp; \mathrm{RON} &amp; \mathrm{MON}\\
\hline
\ce{1,3,5-TMB^{[a]}} &amp; -44.7^{[1]} &amp; 164.65(15)^{[1]} &amp; -5193.1(1.3)^{[2]} &amp; 137^{[3]} &amp; 124^{[3]} \\
\ce{2,2,3,3-TMH^{[b]}} &amp; -54^{[4]} &amp; 160^{[5]} &amp; -5405^{[6]} &amp; 112.8^{[3]} &amp; 92.4^{[3]} \\ 
\hline
\end{array}


[a] 1,3,5-trimethylbenzene, [b] 2,2,3,3-tetramethylhexane


Theoretical predictions of $\mathrm{RON}$

Tareq A. Albahri$^{[7]}$ summarises in his paper the usage of statistics $\mathrm{RON}$ and $\mathrm{MON}$. 


  The research octane number $\mathrm{RON}$, which is representative of the fuel performance during low-speed city driving, is more often reported in the literature than the motor octane number $\mathrm{MON}$, which is representative of the fuel performance during high-speed highway driving.$^{[7]}$


Theoretical intercalculation between the two characteristics is possible.$^{[7]}$ This work was done by Jenkins$^{[8]}$. First predictions of $\mathrm{RON}$ itself were and are still largely empirical$^{[7]}$. These models use regression techniques to assign effective octane numbers to various organic groups. Albahri examplifies the procedure via Anderson, Sharkey, and Walsh$^{[7]\ [9]}$.


  Anderson et al.,$^{[9]}$ for example, developed an empirical model for calculating the RON based on chromatographic analysis of gasoline.$^{[7]}$
  
  In their model, the gasoline is divided into 31 hydrocarbon groups or pseudocomponents, all of which are assigned an “effective” octane number that is estimated by regression of experimental data. The octane number of gasoline is calculated by adding the contribution of octane number[s] from each group.$^{[7]}$


Many similar methods include those proposed by Van Leeuwen et al.$^{[10]}$, Sasano$^{[11]}$, and  Lugo  et  al$^{[12]}$, Ramadhan and Al-Hyali$^{[13]\ [14]}$, Nelson$^{[15]}$, Baird$^{[16]}$,  Twu and Coon$^{[17]\ [18]}$, Rusin et al.$^{[19]}$, Habib$^{[20]}$, Cotterman and Plunkee$^{[21]}.$ Some of these fail to differentiate between isomers, or are valid in specific mixtures.$^{[7]}$ Additionally,


  Although  these  techniques [by Van Leeuwen et al$^{[10]}$, Sasano$^{[11]}$, and  Lugo  et  al$^{[12]}$]  give reasonably accurate results, they are usually too time consuming for planning studies and often the compositional data are not available.$^{[7]}$



Tareq A. Albahri's method of SGC (structural group contribution)


After considerable testing, Albahri found that the best equation to take into account contributions of different groups is$^{[7]}$

\begin{split}
  \mathrm{ON} &amp;= 
  f\left(\sum_{i}\mathrm{\left(ON\right)_i}\right)^{-1}+ a + b\left(\sum_{i}\mathrm{\left(ON\right)_i}\right) \\
  &amp;\quad + c\left(\sum_{i}\mathrm{\left(ON\right)_i}\right)^2 + d\left(\sum_{i}\mathrm{\left(ON\right)_i}\right)^3 + e\left(\sum_{i}\mathrm{\left(ON\right)_i}\right)^4 
\end{split}

where $\mathrm{ON}$ is either $\mathrm{RON}$ or $\mathrm{MON}$, and $\left(ON\right)_i$ is the contribution from the $i$th group. Note how the powers increase from left to right. 

The coefficients still have experimental basis.$^{[7]}$ This equation is a modified version of what Albahri put forward a year earlier.$^{[22]}$ For many compounds the difference between the estimate and experimatal data is just $5$.$^{[7]}$ If the $\left(ON\right)_i$ are known, one only needs the knowledge of the structure of the molecule to apply Albahri's method.

I suggest reading both papers by Albahri since they include tables of data and an example calculation.

Here is a selection of $\left(ON\right)_i$ values.$^{[22]}$


QSPR Model for octane number prediction


In a research article by Al-Fahemi, Albis, Gad, a quantitative structure-property relationship (QSPR) is performed. This includes establishing a correlation relation between $\mathrm{ON}$ and various physical parameters.$^{[23]}$ Again the model is based upon regression rather than derivation from first principles.

\begin{align}
\mathrm{ON} =
    &amp; - (193.53 \pm 319.19) + (1.47 \pm 1.01)M\\ 
    &amp; - (53.06 \pm 31.47)E_H - (8.67 \pm 2.73)B_P\\
    &amp; - (24.94 \pm 19.44)M_R - (50.52 \pm 26.92)\log P\\
    &amp; + (4.33 \pm 3.09)C_P + (3.72 \pm 2.04)C_V + (5.17 \pm 2.08)C_T\\
\end{align}

where 


$M$ is molecular mass,
$E_H$ is hydration energy,
$B_P$ is boing point,
$\log P$ $-$ octanol/water distribution coefficient,
$M_R$ is molar refractivity,
$C_P$, $C_V$, $C_T$ is the critical point (pressure, volume and temperature, respectively).$^{[23]}$


TL; DR

There is no model to date that estimates or derives a result for $\mathrm{ON}$ from first principles. Different models depend more or less on multiparameter regression analysis and experimental data. Nevertheless, the methods of Albahri and others have predictive capability. Often the difference beteen estimated and experimental octane numbers is less than $5$.

Or as Albahri mentions,


  Octane number is one of the most difficult properties to estimate or correlate because of its complex dependency on the molecular structure of the compound.$^{[7]}$
  
  An estimation technique of the octane rating of pure hydrocarbons, though  essential, is nonexistent.$^{[7]}$


So for your second question, I suggest applying Albahri's method because


widest applicability among hydrocarbons,
relatively easy to use,
few $\mathrm{RON}$s are known from experiments,
Al-Fahemi et al.'s procedure requires many parameters which will be a pain to track down (if known at all).


Qualitative considerations$^{[3]\ [7]}$


shorter alkane chain $\ce{-&gt;}$ higher $\mathrm{ON}$ (but higher volatility)
branching of alkane $\ce{-&gt;}$ higher $\mathrm{ON}$
aromaticity in hydrocarbon $\ce{-&gt;}$ higher $\mathrm{ON}$ (but tends to burn sooty with carcinogenic byproducts)
effects on $\mathrm{RON}$ and $\mathrm{MON}$ can be very different


Arithmetic average of $\mathrm{RON}$ and $\mathrm{MON}$ $-$ anti-knock index or $\mathrm{AKI}$ $-$ is also used.

$$\mathrm{AKI} = \frac{\mathrm{RON} + \mathrm{MON}}{2}$$





$[1]$
T.F. Ardyukova, I.K. Korobeinicheva, A.I. Rezvukhin. Atlas of
Spectra of Aromatic and Heterocyclic Substances. $(1973)$, 7. 

$[2]$
W. H. Johnson, E.J. Prosen, F.D. Rossini. 'Heats of combustion and
isomerization of the eight $\ce{C9H12}$ alkylbenzenes', Journal of
Research of NIST, $(1945)$, 35, pp 141$-$146. 
$[3]$ A.
Demirbas, M. A. Balubaid, A. M. Basahel, W. Ahmad, M. H. Sheikh.
'Octane Rating of Gasoline and Octane Booster Additives'. Petroleum
Science and Technology, $(2015)$, 33(11), pp 1190$-$1197. DOI:
10.1080/10916466.2015.1050506.

$[4]$
Jean-Claude Bradley Open Melting Point Dataset. (January 29, 2017)


$[5]$
R. Stenutz. 2,2,3,3-tetramethylhexane. A collection of tables for
chemistry. (January 29, 2017) 

$[6]$
Pure Components Data. (Excel file.) Petroleum Engineering, Texas A&amp;M
University. (January 29, 2017) 
$[7]$ Tareq A. Albahri. 'Structural Group Contribution Method for Predicting the Octane Number of Pure Hydrocarbon Liquids'. Industrial &amp; Engineering Chemistry Research $2003$, 43(24). DOI: 10.1021/ie020306+.
 $[8]$ G. I. Jenkins. 'Calculation of the motor octane number
from the research octane number'. Journal of the Institute of
Petroleum, $1968$, 54, 14. 

$[9]$
Anderson, P. C.; Sharkey, J. M.; Walsh, R. P. 'Calculation of  the 
Research  Octane  Number  of  Motor  Gasoline  from  Gas
Chromatographic Data and a New Approach to Motor Gasoline Control'.
Journal of the Institute of Petroleum, $1972$, 58, 83. 

$[10]$
J. A. Van Leeuwen, R. J. Jonker, R. Gill. 'Octane Number Prediction
Based on Gas Chromatographic Analysis and Nonlinear Regression
Techniques'. Chemometrics and Intelligent Laboratory Systems,
$1994$, 25(2), pp 325$-$340. DOI: 10.1016/0169-7439(94)85051-8. 

 $[11]$  Y. Sasano. 'Measuring Research Octane Number of
Gasoline by Gas Chromatograph'. JP Patent 09138613, $1997$.  
 $[12]$ H. J. Lugo, G. Ragone, J. Zambrano. 'Correlations
between Octane Numbers and Catalytic Cracking Naphtha Composition'.
Industrial &amp; Engineering Chemistry Research, $1999$, 38, 2171. 
 $[13]$ O. M. Ramadhan, E. A. S. Al-Hyali. 'New Experimental and
Theoretical Relation to Determine the Research Octane Number (RON) of
Authentic Aromatic Hydrocarbons that Could be Present in the Gasoline
Fraction'. Petroleum Science and Technology, $1999$, 17, 623.

 $[14]$ O. M. Ramadhan, E. A. S. Al-Hyali. 'Aromatic
Hydrocarbons in Some Iraqi Gasoline and Their Influence in the Value
of the Research Octane Number'. Petroleum Science and Technology,
$1999$, 17, 607. 
 $[15]$ W. L. Nelson. 'Octane Numbers of Naphthas'. Oil and Gas
Journal, $1969$, 67, 122.  
 $[16]$ C. T. Ch Baird. Address: De  la  Haute-Belotte  6,  1222
Vezenaz, Geneva, Switzerland. Unpublished results.  
 $[17]$ C. H. Twu, J. E. Coon. 'Predict Octane Numbers Using a
Generalized Interaction Method'. Hydrocarbon Processing, $1996$, 71,

$[18]$
C. H. Twu, J. E. Coon. 'Estimate Octane Numbers Using an Enhanced
Method'. Hydrocarbon Processing, $1997$, 76, 65.  
 $[19]$ M. H. Rusin, H. S. Chung, J. F. Marshall. 'A
Transformation  Method  for  Calculating  the  Research  and  Motor 
Octane Numbers of Gasoline Blends'. Industrial and Engineering
Chemistry Research. $1981$, 20, 195.  
 $[20]$ E. T. Habib. 'Effect of Catalyst, Feedstock, and
Operating Conditions  on  the  Composition  and  Octane  Number  of 
FCC Gasoline'.  ACS  Symposium  Division  of  Petroleum  Chemistry,
Miami, FL, Sept $1989$.  
 $[21]$ R. L. Cotterman, K. W. Plunkee. 'Effects  of  Gasoline
Composition  on  Octane  Number'.  ACS  Symposium  Division  of
Petroleum Chemistry, Miami, FL, Sept $1989$. 
$[22]$ Tareq A. Albahri. 'Structural Group Contribution Method for Predicting the Octane Number of Pure Hydrocarbons and Their Mixtures'. Fuel Chemistry Division Preprints, $2002$, 47(2), 532
$[23]$ Jabir H.
Al-Fahemi, Nahla A. Albis, Elshafie A. M. Gad. 'QSPR Models for
Octane Number Prediction'. Journal of Theoretical Chemistry,
$2014$, vol 2014, article ID: 520652, 6 pages. DOI: 10.1155/2014/520652.