Does this work well on ball joints, tie rod ends, idler arms, etc.?

Yes it works well on ball joints, tie rod ends, and whatnot. I use it to grease mine. Works great for anything that grease works great for. I use it for many things such as in wheel bearings to lubing my AR15. It's good stuff.

I just bought this grease, but read that it may not be compatible with other greases...Do I need to purge/repack every bearing? Whats the risk?

As stated....always clean old grease from any bearing before you repack bearings. It’s very hard to know which soap base your old grease is and greases with different bases just don’t mix.

Will this grease work well at low temps such as snowmobiles?

Yes. It's all I use on my sleds.

Can this be used to replace Sta-Lube moly graph lithium grease? I can't clean it out of spindles in my mower and want to replace with Red Line.

I am not a mower expert but this is a very good grease if pt in a grease fitting you could put a bunch until it flused the other grease out if it is not inside the engine I think it would be fine. Can you explain spindles? wheel?