If you download a file, you'll see it immediately. No wait times here. It won't show up in your web statistics, however, until the next day, but that's entirely different.
You can, yes. But since the log files sit in your home directory already, I don't see the point unless you want to be able to analyse them yourself?
Fortunately, you don't need to host the domain name at the same place it's registered. If your host will allow you, you can change the nameserver information for the domain name in question, to point to the following:
Then, after the requisit 12-24 hours whilst the new DNS information propogates, your domain name will point people to DH's servers. You can transfer the domain registration itself to DH later, or leave it with your current host–it makes no difference at the end of the day.
Yes, I have an opinion.
Get a minimum 50% off with the "haveadreamyday" promo code, and... have a dreamy day. Original, no?