The most recent dispute has been over the internet being down
They have no legal recourse with that.
Us not responding to a gate being frozen at 1 am
Not really your problem now is it? Is the tenant also going to blame you because his car's lock is frozen?
Us locking the door to the laundry room. This was because they were placing their garbage and using the room as storage. Once we communicated that they were not to do so and not to leave garbage in the common areas, the door has not been locked.
You should have just contacted the tenant in the first place (that's what notices are for). Still, nobody will care about their complaint seeing as they aren't locked out and it sounds like it didn't last for very long.
Us not providing them with a parking space. The lease stipulates that they get a parking space, however we've discovered that the driveway isn't wide enough. We asked them to seek alternative parking, when they were unable to do so...we let them park in the space - we never prevented them from doing so...we just asked to look for alternatives (and would accept the extra cost for such).
Nothing to be concerned about here.
Mold/mildew in the unit. We've never had a problem with other tenants over the previous 2-3 years. However we provided them with a humidity meter, dehydrating crystals, and a dehydrating unit. We also treated the walls and repainted (as the mold growth was minimal). Afterwards when we requested a time to re-enter the unit to re-evaluate - we received no response.
You more than took care of the issue. Tenant has no recourse, especially without having a report from the city that says the mold is an issue.
Complaining about visitors. I've let the landlord know that they are allowed to have visitors. Their main concern is that people are entering and leaving the unit late at night (1 am-5am) and they is concerned for their safety.
It sounds like your mother is living in the building/house in another unit or whatever the setup is. She can't do anything about the tenant's guests, but she has all the power to continue to request that the tenant stops that shit. Everyone is allowed to ask for things. If they're making a lot of noise, that gives your mother more power to make an official request, though.
On that note, the loud music thing could be a way for your mother to have the tenant evicted. If she wants a new tenant, she can call the police (not 911) the next time the tenant is blasting music (assuming it's after 11PM and before whatever AM) and there will now be a record of the tenant causing a disturbance. A few more times and there will be enough to get an eviction. Same goes even if there's no music but yelling/other noise that's audible from outside.
Keep in mind that the tenant is not allowed to withhold rent, no matter what. The tenant has to go to the LTB and can ask to pay the LTB the rent money until the issues are resolved. Since it sounds like the tenant is full of shit, that'll never happen. The LTB will laugh the tenant out of their offices if he comes with those reasons, anyway.
Get a copy of the LTB media that your mother should have given the tenant when he moved in and drop it in the mailbox/slot.