Believe it or not, BOTH are correct. Technically speaking, the person at Lowes is correct. Any inside corners should be caulked. However, so long as your shower floor is tile, it's not attached to the structure in any way. It's "floating" inside a membrane, which means it'll move WITH the walls instead of against them, and a movement joint really shouldn't be needed there, so your installer is correct, too. However, I disagree with him that the corners of the shower should be grouted. Those do need to be caulked.

Yes. Siliconized latex from the same manufacturer as the grout.

My understanding is that you use caulk where there are 2 different surfaces coming together & those surfaces could move. For instance, where the wall meets the floor of a shower or the edge of a tub. Caulk can flex a little bit so if the house settles or you bump a heavy object (or yourself) into the other side of the wall, that little movement would crack a rigid joint (like grout) but won't crack caulk. Grout is more durable & if the wall moves it will move as a single unit, so grout is great there. But where you have 2 planes or 2 independent surfaces coming together, you want to use caulk.

Any change in plane is to be caulk not grout. Grout will not move with the walls expansion/movement, it will crack. TCNA standards are caulk,