This is a hard one to do with small budget or no budget at all. Watching a video implies downloading it, and recording equals uploading. So you are talking about a lot of bandwidth which is not cheap (free will be an impossible utopia really). So you will need to do what every starting business does, which is investing in your tools and resources. If your app’s projected income is higher than the cost of the videos hosting server, then you have a successful business plan. If not, start playing with the numbers until you at least break even. But like I said, videos bandwidth is hard to find cheap, even harder or impossible to find for free.
It depends on each one. Some will let you but I guess they will be more expensive. Usually they don’t allow image or video only traffic. But may be there’s some that do. Google for video hosting options, may be you’ll get lucky.
As an example, this is what Vimeo offer you for $900 a year. I know it sounds like a lot but it’s relatively cheap for all the features they include in that plan. I don’t think it can get much better than this: