Practical advice for calculating optimal prices
A price-response function differs from a demand curve. A price-response function appears in a model where prices are selected by the firm, instead of output. A demand function appears when firms are price takers and cannot influence the price of the product on the market. They will have to sell it at whatever the going rate is.
I will try to answer the questions but they will be very general answers, hopefully they are useful.

The price-response function can be estimated from historical data. I don't know what you mean by transaction data. But if your company is a manufacturer of stuff then transactions could mean the cost of your raw materials and intermediate goods purchases. Then it is possible to estimate a price-response function from this. But it is important to remember that the price-response you estimate is based on your own companies pricing strategy in the past. If your company has made mistakes in it's pricing before, how will estimating this price-response function help that? I think it is better to use a method that takes data from the entire industry instead and benchmark your company against that.

You don't need to do this in very much detail. Just setup your problem generally. Such as, Production Function = Y = F(x1, x2, ...) and cost function = C = G(x1, x2, x3, p1, p2, p3). Solve that problem, what do you end up with? Focus on estimating what is needed in that function in reduced form. I would think what you need is a good measure of your marginal costs.

Estimating the price-response function will certainly have error. But that depends on a lot of things: data availability, model specification, measurement error, etc. You can benchmark against other studies in your field or outside your field instead of trying to removing error completely. You can eventually compare the model's predictions against out-of-sample values.

Unless there is some interdependency between the production of these products within the factories, this should not be a problem I think. For instance, are these products made in the same factories? Does the production of one affect the other? If not, you can get probably get away with solving the problem for each product's price response separately.