Yes, to 'end up' means to reach a certain state or condition at the end of a period of time :  ' if you carry on like this you'll end up getting in to trouble' ;  'after all that effort she ended up failing the exam' ;  'I want to become fluent, but I'll probably end up just being able to patch a few phrases together' ;  and the favourite of a certain type of personal story :  'One thing led to another, and we ended up in bed together'.

Yes, 'wind up' can be used in this way - it suggests failure in an endeavour.  'Wind up' also has other meanings.

I've seen use of "wind up" with the same meaning.