Generally not. Having worked in these sort of scenarios as Security, we have to specifically request a vehicle be towed. Giving a Tow company power to just tow whenever they decided to would be really bad business. It is, however, possible that they do have Maintenance or Security there when closed who would have called for a tow.

As a former tow truck driver, I can assure you it is quite common for an apartment complex to have a patrol contract with their towing provider, allowing the tower to impound any car he sees violating certain specified rules. In my experience, it was most commonly used for fire lane enforcement (since in many areas the complex can be fined for failing to maintain a clear fire lane), but a few complexes also requested that we check permits/stickers.

How do I get it like a tow truck? With tow cables?

That sounds uncomfortable, #10...

That's often the case but most places will make exceptions for a brand new car. Plates and the like come after you buy the car in a lot of places, though I don't know about FL specifically. Either way, OP probably should've figured out the policy vs just assuming parking an unidentifiable car would be fine.

Or just park the new car in a visitor parking space until it has a permit.

How do you even manage that?!