As of July 2017, how big is the monero blockchain download and do you have to download the entire thing?
Blockchain size: Blockchain size on hard drive?
By default - yes. Although, you can use remote nodes to sync the wallet:
There's a way without having to download the blockchain but it comes with a trade-off: you need to give your viewkey to some service and then it does the scanning for you.
Centralization of users around such services comes with some risk:
That's why the plan is to bake-in the backend to the official release, and enable anyone to run his own back-end (like, you let your PC be synced, and hook your mobile to it).