Hi @thetinus,
first of, two advice:
http://www.wpbeginner.com/beginners-guide/why-you-should-never-upload-a-video-to-wordpress/
http://support.hostgator.com/articles/specialized-help/technical/wordpress/why-videos-should-never-be-uploaded-to-wordpress
To your question a) & b)
yes you can code a custom (user) profile page. And yes you can use a player on it.
But all depends of the number of videos attributed to each user, how many user will use this, the config of your server and of course your personnal knowledge about templating BP and php coding.
Also, embeding hosted videos on profiles from official WP providers is very very easy – and powerfull. Doing the same from your own server is a bit more delicate to handle.
https://codex.wordpress.org/Embeds
Codex is your friend.
BuddyPress Theme Development
BuddyPress Plugin Development
If you have questions, please ask only one by topic and try to be very specific and detailed.
what you want to do
what you already did – give example or pastebin your code
and what you think should work and doesn’t.
thank you !