vamp... read the Newbie FAQ (bottom of screen), the HELP page (see the button at the top of the screen near LOGOUT) and the Welcome Newbies thread (at the top of this section.)
You can earn more than 1 skill point. The frequency that they come at is based on your XP. eg When you're at 50XP they come 1 every 10-20 XP. When you're on 1,000,000 XP they come one every 48,00XP or so.
Yes you can get more money for solving cases. You can also earn money by doing the 'favour' cases and selling the reward equipment you get from them.
If you get 3 false accusations you have 2 choices:
1/ resurrect/reactivate via the Log In page as Jas says, though there are drawbacks to this at your early age, including starting in the negatives second time round
2/ start a brand new detective if you're not too far in to this one, if you want.
You can also go visit Shady in your current state (he's in the Bar) and pay him off to have your two false accusations quashed. The more XP you have the more he costs.
It'd be quite cheap for you right now.