Can I do plank exercises in my second and third trimester?
We can do modified plank work! The reason I don't have my prenatal clients do regular plank work in their second and third trimester is because it causes pressure in the abdominal wall, which contributes to Diastasis Recti (the separation of your abs).

In the second and third trimesters, could I modify abs on back by raising my back (on forearms, ball, wedge) and just doing the leg movements?
Technically, yes, you can do this as long as your implementing 360 breathing and your pelvic floor/not coning, however I tend to advise avoiding abs on back in their second and third trimesters. I think doing my pelvic floor and 360 breathing exercises solely in those sections is going to be more of an efficient core workout than modifying abs on back on your forearms.

Is it safe to do bridge work in my second and third trimesters?
We don’t want to be on our backs for long periods of time, but one song of bridge work is safe to do in our second and third trimesters. However, if you feel at all lightheaded during bridges then I would recommend skipping that part of class.

Is it safe to do core work during my pregnancy?
As far as when you should stop doing abs on back, you can continue working out the same in your first trimester as you were prior to becoming pregnant. With abs on back, if anything feels off, modify and/or slow down. If it still feels okay during your first trimester you can continue to do it. In your second and third trimester you should not be doing any abs on back, you can do elevated or modified plank work.