Hacking Vehicles in Watch Dogs

This is happening to a lot of other people, across all platforms. The official Ubisoft forums have a 150-page thread on the issue.

So, Ubisoft is aware of the issue, and is (presumably) working on a fix.

[Edit] The latest word from the Ubisoft staff:

(June 1st 2014)
The team are working on the issue, is the best I can say at the moment.

It appears to happen when you unlock in-game items from the Ubisoft store. Unfortunately, there is currently no known workaround. You'll either have to start a new game, or wait for Ubisoft to provide a fix.

And in the meanwhile, don't purchase any in-game items from the Ubisoft store!

No, unfortunately you can neither repair nor modify your vehicle. There is basically no need to repair vehicles, because you can simply "respawn" any vehicle you could have destroyed with the "Car on Demand" app. I have completed about half of the game and have not come across anything to modify a vehicle, but feel free to correct me if something comes up later.

As Lyrical already mentioned in a comment, you unlock new cars for the app by simply entering them. If it is the first time sitting in the type of vehicle, the message "new vehicle unlocked" will appear in the top right corner and then you can buy access to it in the app. Another way to get cars (often rarer ones) is to hack civilians' phones. To make this easier, you can unlock a skill in the "Hacking" branch of the skill tree that marks civilians with especially valuable hacking opportunities (including cars, songs, crimes and rich bank accounts) with a blue square, so you can already see them from afar. The chance of finding vehicles that way is still rare though. Hacking people's phones is one of the few things you can do while waiting for a multiplayer session to start, so that is a good time to go searching for people with cars.

A good way to unlock sports cars, is to play Fixer Contracts that require to race through the city with a given car (which was already a sports car in the first mission I played). You can unlock new fixer contracts by hacking ctOS towers, which reveals new missions and points of interest in the area.

If you want a specific car from these missions, here is a list by segmentnext.com with fixer contracts and the vehicles you will be driving there.