Also, if you take into account that all components change their nominal value in temperature, so the fact that it depends on beta, and you resisitors and BJT changes in temperature it makes it floating. What I mean is that during normal funcitoning it will get hotter and that will change the output you calculated.

Agreed in those 2msec he will have 45W in there. Datasheet says it can handle that, but additional power disipation is still required. :)

even though there are people who have this knowledge you are asking too much. Take a book from the library and read chapter 1 about charges. Many of these questions will be answered.

If I were you I would start by making the configuration with first and second input(your second and third paragraph). You only need gates and multiplexers. The third step is more interesting as you have to modify a little bit the alredy existing circuit.

@OlinLathrop is right. Many manufacturers do this to prevent people from using their products in other applications.

Also don't forget that there are institutions which function is to check this kind of products.

I'll add you a simple circuit now with a simulation results.

Question. Do you turn off the source of sinuoidal signal after modifying the frequency? From your description I understand that in the begining you have 2 Sinusoidal signal in SYNC but one if shifted. What i try to say is that is seems that you lose SYNC of the input signals. Could you check it with scope. Also you could use INA128 instead of OPA2111 + INA105.