I have an old single 2.4 ghz router. If I upgrade to a new high speed router will I get better streaming video?

You certainly should. Improves stability as well.

It depends on what's rate limiting - if it's your wifi maybe, if your internet connection is slow the router won't make a huge difference though some products are better with poor lines.

Can you hook up a laptop to the router via ethernet and test the internet speed via speedtest.net?