Domain names may be registered and hosting arranged with another providor.
1&1 are one of the least expensive domain registrars, and they have the added benefit of full name server access via the control panel, which other registrars often leave out or charge extra for.
This means you can register a domain with 1&1 and then host it with another web host if you wish.
Having said that, 1&1 also offer hosting, so it may make sense to register the domain and host it with the same company at the same time.
Estimating web space requirements is difficult without knowing all the angles.
Web pages are tiny text code files, so hundreds of pages will only amount to a couple of megabytes. One short video could well be into the tens of megabytes, depending on the resolution and sample rate you save the file at.
I can't really think of many reasons why I'd want to download video promotional material to my phone, but that's possibly only me. I'm an old fashioned soul at heart and my phone is for making and receiving calls, and occasionally (and reluctantly) texting.
Any host offering your site more than 10 or 20 megabytes of space will be more than enough for pretty much anything you want to throw at it.
If mobile phones are your intended video audience though, pay very close attention to the file formats, resolution and file sizes or you will quickly alienate your downloaders.