First of all, thank you for taking the time to answer me, it's appreciated, and sorry for the late reply!
I'd say I usually chop a tree a good distance from my cars because they sometimes glitch, and yeah, i usually chop the tree where the trunk and branches meet because it's faster (I mean, when I cut the trunk at that specific place, all the branches fall) overall. I haven't noticed my cars move or glitch, because like said earlier, a good distance away from them.
The times I've died, the servers were fine and didn't lag, and I haven't tried using a hardened axe. Never chopped a tree while it was in a vehicle.
Once again, thank you for answering and I apologize for the late reply!