Running linear regression with 3 predictors and term interaction [closed]

I'm trying to ascertain which portions of my social networking web app are driving user retention (so I can fine tune my site).

Step 1 (in my view) is to do an inference on what predictors are correlated to user retention. I'm taking the linear regression route for this.

Consider that my web app has 3 sections: A1, A2 and A3. Roughly speaking, one can start by testing the hypothesis whether engagement in any of these sections determines user retention. I.e. one can say:

Y^ = B0 + B1*A1 + B2*A2 + B3*A3


But what if there's interaction (or synergy) between these sections? As in all three sections feed the other two as well, creating a multiplier effect.

In that case, how would I model the regression? Would it be something like:

Y^ = B0 + B1*A1 + B2*A2 + B3*A31+ B4*A1*A2*A3


Secondly, wouldn't R^2 go up anyway as I add more and more predictors? For instance, imagine section A3 of my website is actually a dud. Would this fact appear in the coefficients once I run the inference (given I'm also including interaction terms)? Or in other words, how to play around with my model to remove weak predictors?

Sorry for the noob question, I'm getting back into statistics after a long hiatus. Would love a layman's explanation here.