1. "A more sim-like driving feel than arcade."
For a relatively hardcore sim, the contenders would be rFactor 1/2, the pre-Raceroom Racing Experience Simbin games (GTR, GTR2, GTL, Race 07), Game Stock Car, iRacing, Assetto Corsa, Netkar Pro and Richard Burns Rally from what I can think of.
2. "Career mode where you can win prize money to buy new cars and parts."
This one's tricky. Long story short, this is a feature most sim games simply don't have. It's a bit like Microsoft Flight Simulator, you just jump in and go with whatever you want. In saying that, these are the games that have anything resembling a career mode:
- GT Legends (GTL) has a career mode, where you move up the championships/rankings and either buy or win cars along the way (can't remember).
- Richard Burns Rally (RBR) has a career mode where you win cars and unlock rallies for use in single stages later on.
- Assetto Corsa is said to be getting a career mode in the future, but any details of any such mode are yet to be released.
- iRacing has a career mode, but it's a bit different. All the racing is online, and you pay for your subscription, cars and tracks with real money. I know it's not what you've had in mind but I thought I'd mention it anyway.
Apart from these, most sims also have the ability to make custom championships (either in the game or with external file creation/editing) - not a career mode but enough for me.
Regarding buying parts - rFactor 1 (don't know about 2) has the ability to "upgrade" the car using in game money, but most mods have prices at $0 or $-1 and use the upgrades mostly for non-performance stuff (e.g. visual stuff, type of gearbox simulated i.e. sequential or H-Pattern).
3. "Support for Logitech G27."
Any proper simulator will support this (including all those mentioned above), it's one of (if not the) most popular wheels out there. You shouldn't have any issues here.
4. "Cockpit view would be nice but is no must have."
Same as above, any proper simulator I've mentioned so far has it.
5. "Car tuning (suspension, gears etc.) would be nice but no must have."
Once again, this is something all of the sims I've mentioned have and it is a feature.