If Cersei had turned out to be infertile, would Robert aim to remarry or legitimise Edric Storm? Also what would Tywin and Cersei have done in this situation?

That’s a good question.

I think the politics would get very complicated. Siring an heir is a vitally important task of a monarch, especially a monarch of a very young dynasty – to quote Hillary Mantel, “if a king cannot have a son…it matters not what else he can do. The victories, the spoils of victory, the just laws he makes, the famous courts he holds, these are as nothing” – so there would be pressure on Robert to have a son, which would mean putting Cersei aside.

On the other hand, there would also be counter-pressures: Robert has two brothers to stand as heirs, which means that the dynasty as a whole is in less danger than his own personal line is and that both Stannis and Renly have an incentive for Robert not to sire an heir; House Lannister is politically important in the broader political coalition that supports the new regime, so setting them aside is potentially dangerous.

The big question that would need to be answered here is: if not Cersei, who? This is an area where @goodqueenaly is more knowledgeable than I am, so I’ll toss this over to her.