The stock carrier can't hold the weight of bigger tires without damaging the hinges, so alot of people just ditch the tire carrier for that reason. Many upgrade to a tire carrier/ bumper further done the road though, but the nice ones tend to be pretty pricey.
Well I don't think it is a good idea but some put the spare inside if they take out the back seat. If some oversize tires don't fit they just don't run spares. I personally would never go anywhere without my spare which is a matching wheel and tire to the other 4. I just make the needed mods to run the large tire and go with it. A lot of people will get after market tire carriers and bumpers but I did not want the extra cost and weight.
I dont think the tailgate can support much bigger tires than stock, weight and sizewise. Also if you have positrack differentials, your spare has to be the same size as the tires you run or you mess up the diff.
I had to take mine off because the stress caused by the stock tire + bike rack and bikes wore out the hinges. I'll be putting it back on once I get new tires since I put new hinges on but will never use a tire mounted bike rack. I'd love to get a bumper mounted tire carrier, but I'll have to save up after I do other stuff (another lift)
If you keep your tire in the back, I hope you keep it strapped down. A tire/wheel will hurt a whole lot in a collision flying through the compartment.