I recently customized my standard car for Franklin (Dodge Charger, don't know what it's called in GTA) and spent a lot of money on it. I saved it in the garage and I'm afraid to take it out and go do missions and lose it. Will these customizations be lost if I lose the car at any point or will it be saved when the car respwans at the garage later on?
Each character has a personal unique car. If lost, it will appear near the garage of his safehouse. Sometimes, it will be in the vicinity of the character, when you switch to him.
Any customizations made to it will be permanent and cannot be undone, unless you re-customize it.
At some point in the story...
Jimmy De Santa will "borrow" his father's personal car, at which point it will become unavailable until Jimmy brings it back.
Michael's car will be customized, so you might want to customize his car only after that event.
No matter what you do, you cannot permanently lose your car, and any and all customizations done to it is permanent. Do not be afraid to abuse your tuned personal car as a cannonball, and don't be afraid of losing it.
Basically, tuning your personal car is a safer investment than tuning any other car.
Please take a look at this questions and its answers: How does the garage system work in GTA V?
If you customize a standard car, you have to at least once save it in a garage (the big ones, not the one next to your save house). If it gets destroyed during a mission or while cruising through the city, it will then re-appear in the garage.
But if it is stolen or abandoned (or replaced by another vehicle during a mission cut-scene), it gets impounded. You have the chance to get it back for a fine, but there is only one slot for impounded vehicles, so with the next car you loose, your first one will be lost forever.
I'm not entirely sure where to get the impounded vehicle. (Probably the parking lot next to the police station where the tow truck missions start.) I'm also not 100% sure if the impound-ability gets reset when repossessing the car, when it's saved in the garage again or even never at all.
Also there are reportedly some glitches happening. So your car may as well vanish from your garage, may be replaced by another car in the garage, may never be impounded or not saved after repossessing it. Good luck!