The most popular Pokemon challenge is probably ye olde standard Nuzlocke (can only catch first Pokemon on route after obtaining Pokeballs, must release/permanently deposit anything that faints). The other one I can think of off the top of my head that would apply to Red/Blue is the monotype challenge - using Pokemon of only one type. I've tried this with Water before, but I've never finished it (not because of difficulty though - I just stopped playing the run at some point and didn't resume).
You could also mix and match (e.g. restrict yourself to one type of Pokemon and release things that faint).
I don't know about Nuzlocke's rules, but monotype allows the use of other types for HM purposes, but not for battle.
One major difference as far as the workload of streaming vs. recording-video-but-not-streaming is that in streaming you can get away with not editing the footage (b/c that wouldn't really be practical). Of course, maybe you don't think editing is necessary anyway and so you don't do any. Some people will do streams on Twitch and then they'll upload the footage afterwards to YouTube - most seem to just upload the raw stream footage, and most people don't seem to complain about that. Some will make the Twitch chat visible as well in the YouTube footage.
I'm not sure how any of it works, though.
I was about to say that
anyway Nuzlocke rules
-When a pokemon's hp reaches zero, The pokemon is dead, you must release em or permabox em
-One pokemon per route, failure to do so and you must wait till you find a new route
-Name the pokemon, its your best friend until they die
Addition rules
to rule 2
Doops Clause
If you have caught that pokemon before, you can running til you find something new (Includes evolutions)
Shiny Clause
This doesn't apply to Red/blue/yellow so if you want shinies in kanto play Fire red or Leaf green, on the off chance you find a shiny you can catch it and or use it
General
Set style only
No switching
Level Limit
You can't go higher than the gym leader's strongest pokemon, the level limit rises when a gym leader is defeated, highest is the champions strongest pokemon