Should this wheel be fixed or replaced? So, I have this tire with a factory sealed bearing.  It has a few broken spokes.  TO get these off, I'd need to remove the freewheel.  Sadly, the "studs" on both sides seem to turn independently, suggesting the axle is broken.  As a result, I don't know how to brace the axle so I can unscrew the freewheel and get it off.  Or should I even ride it like that at all?

What I'm asking is, is this wheel repairable, or should I buy a new one?
Your freewheel seems busted from what you say, it happens when the cogs are worn, nothing much to do there.  

Everything is reparable but it sometimes takes a lot of time for not a lot of gain. Question is what is the value of your wheelcircle and from picture i'd say that's and old aluminium one. I don't know how that is where you live but in my neighborhood you can get a "brand new" used one for under 5 bucks. If the freewheel is busted you have to replace the entire bearing, meaning you have to take the entire wheel apart and then do the spoking from scratch. If you have a bit of experience and tools that is a good 1-2 hours of work, if it's your fist time count merely your sunday afternoon on it.

So my conclusion is : get a new wheel and recycle this one into an artistic project of some sort :)