I usually take the giver's wishes into account. But I think once a child is old enough to know what presents/money are, it is a bit cruel to say - "hey here's some money but you can't have it because you must put it into a bank account." A gift is a gift and the child should be able to spend it if they want. It is a bit different if they are doing little jobs to earn money say, rather than a pure gift. Essentially, the parents have the say. As a gift giver, I have given money to my nephew and I don't really care what he does with it. I expect he will spend it on some crap but at least it is crap he wants rather than crap he doesn't want lol. I think it would be a bit rude to tell him he must bank it instead.

And to your final question - yes or at least, request bank account details so they can transfer the money over themselves without making you go bank it.