Should this wheel be fixed or replaced?
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 :)