Ejant- Nice pic of the rabbit :xyxthumbs
samiam513 - I'm testing out a new (used) camera and thus sorta photographically-oriented at the moment, so I'll throw out my $0.02.
First, remember that the smaller the aperture (MORE depth of field) the LARGER the f-stop NUMBER (aka "stopping down") and the LESS light getting to your film. Same as saying the larger the aperture (LESS depth of field) the SMALLER the f-stop NUMBER (aka "opening up") and the MORE light getting to the film.
1. Generally if you "stop down" (bigger f-stop NUMBER- yes, this IS confusing, huh?) to increase the depth of field you're more likely to UNDER expose. Gotta have the right exposure (aperture and shutter speed combo) to get light/dark bit right. The "sunny 16" IS a good rule of thumb (and very handy when your meter dies

), but you still often have to adjust for light/dark subjects and you gotta watch for shadows, etc. It's definitely worth studying up on this! Little diffs in exposure and focus often make the difference between ~OK snapshots and pics that people rave about.
2. Yep, as Ejant said, make sure you get the girl's face right (for a number of reasons

).
2 & 3. Lenses differ, but there are GENERAL rules-of-thumb for what f-stop=what depth of field for different focal lengths. If they aren't marked on your lense (a scale with f-stop numbers repeated on each side of a central indicator), check your manual.
I usually try to get BOTH the girl and the car in focus. One method is to focus on the girl, check the distance on the lens's distance-scale (say, 10'). Then focus on the part of the car that's farthest away from you and check ITS distance (say maybe 15'). Set the focus for the middle of the two distances ( here 12.5') and use an f-stop that will keep the whole 5' (from 10' to 15') in focus (maybe f8). It might be a good idea to err on the side of caution and pick an aperture that'll give you a bit MORE depth of field than you think you need (like f8.5 or even f11) because if you mess it up you might end up with NEITHER the girl OR the car in focus

I'm noticing that this works a LOT better with some cameras/lenses than with others so maybe it'll work for you and maybe it won't.
4. For pics of bugs you can get close to, you might want to consider a lens with "macro" (very close-up) focusing ability. But yeah, can't beat a tele for wildlife, as evidenced by Ejant's rabbit pic. A tele with a broad range (like Ejant's 80-300) would be great.