I would like to install a sink in my basement but the only drain that is close to it is being used for the HVAC. Can I use this existing drain for the sink as well?
You have to ask your local building inspectors. Some places allow grey water drainage into their overflow drain but most don't.
One of the problems you will have is if your sewer backs up it will back up into your sink drain. This isn't a huge deal but could get messy. Also you would be require to secure the PVC to the floor in some way.
I personally don't see an issue with it. There is no real downside other than it looking slightly ghetto. Most inspectors would just tell owner to set it up after they pass inspection - hence most cities won't recognize it as finished plumbing. Will it work though, sure as long as you are not sending a ton of mud or crap down your sink.
Floor drains often go to small sumps for diffusion into the soil, since they're just intended to handle occasional spills. For more volume, or for anything you wouldn't want your back-yard garden soaking up, you really want a proper drain connection.
(The pipe is for condensate, which is fairly clean and fairly small amounts.)
No. You cannot do that. It is against all rules of common sense, the national building codes of Canada and the US, and almost surely against the local plumbing ordinances you have. A sink needs to be connected to your sanitary sewer. The floor drain will be connected to your storm sewer. If you are in an older home, where they are shared, you could do it, but you will have to excavate the floor and make the connection sub-slab with rigid connections and not flexible hose or pipe. If you don't do it properly, you could end up having a flood in the basement, or having sewage or storm water or both coming into the basement and not having any way out.