Relationship between Gibbs free energy, enthalpy, entropy and rate of reactions
I'm having a little trouble constructing the bigger picture of chemical reactions.


Let's go step by step.


  From the collision theory we know that molecules react if: 1) they collide 2) they collide bearing a sufficient amount of energy to surmount the activation energy and collide at a specific geometric orientation. 


Yes, in order to react, molecules taking part in a (bimolecular) reaction need to:


be in contact;
be correctly aligned to each other (a random alignment may not react);
have enough energy to overcome the activation barrier ($\Delta G^\neq$).



  From the relationship between Gibbs free energy, enthalpy, entropy and temperature we know: $\Delta G = \Delta H - T\,\Delta S$,
  and that a reaction is only thermodynamically favored when $\Delta G &lt; 0$. And even when $\Delta G &lt; 0$ the reaction may still not occur at a measurable rate because it has too high an activation energy.


What you're describing is the Gibbs reaction energy: the Gibbs energy between products and reactants.
This indeed tells us how thermodynamically favourably the reaction is ($\Delta G &lt; 0$).

But the phrase in bold above is wrong.
The reason is that the Gibbs activation energy ($\Delta G^\neq$) is the Gibbs energy difference between transition state and reactants.

As Mithoron mentioned, Gibbs reaction energy would give you the equilibrium constant for the reaction:

$$K = e^{-\tfrac{\Delta G}{R T}}$$


  From the Arrhenius equation we know: $k = A ~\mathrm e^{-E_\mathrm a / (RT)}$


Generally, from Eyring theory,

$$k = \frac{k_B T}{h c_0} e^{-\tfrac{\Delta G^\neq}{R T}}$$

Comparing to Arrhenius equation, see:

$$k = \frac{k_B T}{h c_0} e^{-\tfrac{\Delta H^\neq - T \Delta S^\neq}{R T}}
= \frac{k_B T}{h c_0} e^{\tfrac{\Delta S^\neq}{R}} e^{-\tfrac{\Delta H^\neq}{R T}}$$

Thus,

$$A = \frac{k_B T}{h c_0} e^{\tfrac{\Delta S^\neq}{R}} \\
E_a = \Delta H^\neq$$

That's what you would find from an Arrhenius's plot.


  Linking all these together:
  1. Reactions only occur when particles can collide with each other.
  2. Colliding particles will only react if they bear sufficient energy (can be increased through temperature and/or adding other external energy sources) and have correct orientation (chances can't be increased?).
  3. Only particles in thermodynamically favored reactions have enough energy to overcome $E_\mathrm a$ and so will spontaneously occur without external energy applied. But they may occur extremely slowly if they have a very high $E_\mathrm a$ (according to the Arrhenius equation).
  
  Is this reasoning correct? 


The reasoning is correct, except for point three above (in bold), for which the same error is found as before: reactions may have a big $K$ (small $\Delta G$) but small $k$ (big $\Delta G^\neq$) or vice-versa.