At some point, my heroes will all be maxed and my antiserum at warehouse capacity. Does this mean ..

I just keep resetting one of my four star heroes and use the antiserum gathered from the lairs to upgrade them. Costs 150 diamonds so not expensive. This puts the antiserum from the reset in your bag, so the only thing you lose will be the diamonds. Accumulating the antiserum this way also allows you to upgrade any new 5 star heroes you might get later really quickly. Also comes in handy for upgrades above lvl 350 which take a lot of antiserum.

Keep upgrading your buildings i.e. your warehouse. Upgrading warehouse increases its capacity and you’ll be able to keep/collect the antiserum.

Yes, you will definitely be able to continue making joining raids. It’s just that literally any task you do that would earn antiserum as a reward will keep bumping your antiserum unless your warehouse capacity increases to accommodate 😃)

You will be able to hunt zombies and do any other activity that rewards antiserum. you just don't get the antiserum.

If you need to upgrade heroes for the quest then you can reset a hero to clear it back to level 1 and upgrade it again.