Enzyme kinetics at the chemical level
The answer to the first part of your question is that we don't take the initial 10% of a progress curve (velocity vs time) as a measure of activity, but we measure the initial rate of the reaction.  We do this by drawing a tangent at the origin.  

It is merely a 'rule of thumb' that progress curves are practically linear provided that not more than 10% of substrate has been used up. If there is significant slowing down before 10% is used up - and there may be if the substrate concentration is way below Km value - then it is not valid to draw a line through the points: we must draw the tangent to the curve. That is, it is not acceptable to take the average rate during the first 10% of the progress curve. 

For this reason it is desirable to use as sensitive a method as possible when measuring an enzymic rate, and to never rely on a single time point (as is common in radioactive assays) without sufficient controls to justify that it it is the initial rate, and not the average rate (or something else) that is being measured. 

The reason we measure initial rates is that is makes things simpler on two fronts.  

Firstly, the decrease in rate with time of an enzymic reaction may be due to many factors: product may be inhibiting the reaction, the reverse reaction may become important, the enzyme or substrate may be unstable. It is only during the initial rate period that conditions are accurately known, and such  effects may be legitimately ignored. It was Michaelis and Menten that first realized the importance of measuring initial rates, and of the great simplification principle that ensued, in enzyme kinetics. (see Cornish-Bowden, 2004). 

Secondly, the rate laws themselves are greatly simplified.  Provided that eo &lt;&lt; So (the initial enzyme concentration is very much less than the initial substrate concentration), we may take the So as being equal to the substrate concentration and 'plug in' this value into our rate law (such as the Michaelis-Menten equation), and ignore the effect of time on the value of S.  In addtion, we can set all product terms to zero. This can greatly simplify things.  

For example, the rate law for a reversible single-substrate mechanism (see here) is the following: 

$$ v = {
  {{{V_{max}^f}\over{K_{m}^s}}\ S\ -{{V_{max}^r}\over{K_{m}^p}}\ P }\over{1 + {{S}\over{K_{m}^s}} + {{P}\over{K_{m}^p}}}}\ \ \ \ \ (1)$$ 

This one is not as complicated as it looks: we have just defined a Km and Vmax for the forward and reverse directions. But let's set product to zero:

$$ v_i = {
  {{{V_{max}^f}\over{K_{m}^s}}\ S_o\ }\over{1 + {{S_o}\over{K_{m}^s}} }}\ \ \ \ \ = \ \ \ {{V_{max}^f S_o}\over{{K_{m}^s} + S_o}}\ \ \ \ (2)$$ 

We now obtain the familiar Michaelis-Menten Equation (where $v_i$ is the intial velocity). 

The second part of your question is more fundamental. I'll take it to mean the following. Why does an enzymic reaction not follow second-order kinetics at all substrate concentrations? Why doesn't doubling the substrate always double the rate (as is often the case in chemical kinetics)? A. J. Brown in 1902 suggested that the reason for this is the formation of an enzyme-substrate complex and that at sufficiently high substrate concentration all the enzyme would exist in such a complex and the enzyme would become 'saturated'. The formation of an enzyme-substrate complex, of course,  is now a cornerstone of enzyme kinetics. One of the early key pieces of evidence for an enzyme-substrate complex was the observation that an enzyme is much more heat-stable in the presence of its substrate than in its absence. 

Finally, it is probably worth pointing out that some enzyme operate way below Km values and at 'physiological' concentration of substrates do obey (to all intents and purposes) second-order kinetics. Catalase, for example, has a Km for hydrogen peroxide of about 1 Molar (Ogura, 1955), and one assumes that physiological concentrations never reach this level. (We need to be a little careful about catalase, however: it is one of the few enzymes that is inactivated by substrate). 

A great reference on enzyme assays is the following:


Principles of enzyme assay and kinetic studies by K. F. Tipton in Enzyme Assays: A practical approach


For an introduction to enzyme kinetics, and the early history of enzymology, I like the following:


Cornish-Bowden (2004) Fundamentals of Enzyme Kinetics 3rd Edn. Portland Press, London.