In q theory, a firm uses investment to maximize the risk-neutral expectation of the integral of exp(-r*t)*(operating profit(t) - investment costs(t))dt.

Why does the firm use the risk-neutral measure and not real life probabilities? Because it otherwise needs to discount future cash flows at the SDF instead of exp(-r*t) and as q theory is a partial equilibrium model, the SDF is exogenous?

Two separate problems. Marginal Q is always a sufficient statistics for investment, no matter whether you derive it using risk-neutral or physical probabilities. Roughly what you are saying is that the firm has an arbitrage opportunity to issue equity (or debt) whenever is mispriced. They should do it state by state every time they can make money for free. This is the reason why Q models are a failure: they always predict excess trading for a firm.

Regarding risk neutral expectations. Why should a firm be risk averse? I would expect the manager to be risk averse, but in that case you need to put an extra layer of agency in your model.