Mathematical framework for modelling the relationship between price and sales of a product
The main problem of your framework is that you talk about sales but model only demand. Assuming that the supplier is not a monopolist you should take competing suppliers into account, when estimating sales. To do that any real world examples will have to define the extent of the market. Having said that your framework is a pretty good job as a first try! 

@3, 4. Classic micro relies on the irrelevance of independent alternatives. Translating this to your framework $a_2 = f(P_2)$, unless $P_1$ directly affects the consumer's budget in $t_2$. For example - if oxygen was more expencive than your income yesterday, you had to borrow to buy it and today will have to repay your debt. As far as such intertemporal concerns may be important, you need to specify the consumer's planning horizon, which will generally depend on the type of good. For example - if durable goods are involved - cars, fridges etc., and I buy today I won't return as a consumer to this market for the next five years no matter what. However if I know (or expect) that the price of cars will come down next month I will wait until it does and buy than. I.e. it can be that $a_1 = f(P_1, P_2)$. I cannot see why $a_1$ should affect $a_2$ or vice versa through a channel different than the price vector unless there are some asymmetric capacity issues in some periods and demand in excess of capacity thus moves back and forth between periods. I am not exactly sure about search frictions but I think that their effect is quite similar with regard to your question. As long as you have simple consumer goods that you buy every month $a_t = f(P_t)$ is okay.

@5 If the oxygen is so expencive that the purchase violates your consumer's life-time budget constraint I am sorry but he is going to have to die. In this case it is obvious that $a_2 = f(P_2)$ only.

@8 There are goods for which the quantity demanded increases when their price goes up. They are called Giffen goods. Apart of that if your consumer from @5. can afford oxygen he will buy. I.e. it can be that $P_1 &gt; P_2$ and $f(P_1) = f(P_2) = 1$. 

Demand is usually modeled as a $Q_t = f(P_t$, control variables) + $\epsilon_t$,

where $f$ is deterministic and all randomness comes from the error term, which should be familiar to you from your daily work.