Which GTA Online cars can you steal and own?

You can still upgrade the Sports cars like the Inferno, Banshee, etc. However, you can not upgrade them if you stole it. By LS Customs telling you the car is "too hot" it means that it's a Super Sports care that was stolen. To upgrade one of the vehicles, you have to actually spend your money on the web and purchase them. No worries though, buy insurance and a tracker for them and you will always be able to get your vehicle back even if it blows up.

Does anyone know which cars can be made your "personal car" by stealing? Or which ones can't?

There is a list of cars that fit this category in this GTA wiki article: Los Santos Customs/Vehicle Prices and in the answers to the Arqade question: Which stolen cars can you bring to LS Customs and what are their sell prices in GTA Online?

Out of them, is there a "best" non purchased car in terms of speed/acceleration/the usual?

According to Broughy, who tests vehicle top speed by driving them on a road (as opposed to just getting the speed stat in the game files), the current fastest vehicle that can be made a personal vehicle by stealing is the Pegassi Ruffian (a motorcycle), with the Vapid Dominator second, and the Maibatsu Sanchez (a motorcycle) third. Among cars, the Vapid Dominator is the fastest with the Karin Futo coming in second.

Putting the stats of those vehicles here for better comparison:

Sources:

• Broughy - GTA 5 Vehicle Info, Performance & Tier Lists
• Broughy's top speed testing methodology
• Broughy's lap time testing methodology
• GTABase.com GTA V Vehicle Database