Use a web wallet. Problem solved.
Not for web wallet. With exponentially growing blockchain size, it will become a problem for them very soon.
Of course not: a web wallet would host only a *single* copy of the blockchain for all its users, not one for each of them.
Of course yes. They will need to double their storage every several month. Do not think that it will be for free.
I have visions of sysadmins frantically purchasing new drive after new drive to add to their servers, all sitting empty.
Hint: with the current block size limit, a 1 TB drive will die of old age long before it fills up in 2032 (at the soonest).