I'm looking for knowledge!!!

STORY #1
OK OK OK, Here's the story. I was looking out of the window at the feeder (which is like 50ft from the window ), and Underneath me I see a bird sitting on our satellite dish. It quickly flew away (not that it saw me), under the stairs that leads out of our back door.

Under the stairs of our backdoor, I've spotted birds (I believe it was just a titmouse) flying around there. Do you think there could be a nest there? I've found lots of leaves, STRING, some hay-like stuff. Now I today after I saw that bird I decided to check out underneath there again.

So I was walking towards there, and right outside of it, I found a dead mouse (or a rat, not sure. I'll post a picture if you want me to)

I've never seen a predator lurking around in my backyard, some do fly over head but never really come back. But this mouse/rat was ontouched, no blood showing, just laying there wideways. I didn't bother to touch it..but what should I do, or what could this mean?

STORY #2
So I've been browsing the forums and decided I have a new goal. That's to get a bird to eat a seed from my hand! Yay? No. The titmice and nuthatches let me get super close, they trust me. When I see them land on the magic tree (where my feeder is hanging from), I stick my hand out slowly with the black oiled sunflower seeds in my hand. They just look around and without going to the feeder or my hand they fly away. One of them went to the feeder, didn't get any food and flew away like it was scared. Any tips on gaining its trust (They've known me for over a month now, every weekend I see them)

Story #3
So there's this downy woodpecker (male). Normally I am able to get close to him (as close as the titmice), but recently he's been really shy. He doesn't even look at me but when I get fairly close (like 12 ft) he ends up flying away and I don't get a shot of him!

Story Oh god make these stories stop #4
I've noticed some mourning doves I believe scattering around below the feeder, cleaning up the remains which makes me happy that nothing is being wasted. The problem is, all the birds are at the feeder at the same time. So where there are doves and stuff around, there are finches and stuff. When I get withing like 20 feet of these guys, the doves go CRAZY flying away making so much noise, thus scaring away all the other birds! I don't think there may be much to say about that, but if you do have anything in mind, say it!

I'd be suspicious of a dead rat or mouse with no injuries just laying out in the open. Make sure nobody put poison out and throw it in the trash.

All the pics I've taken of birds in the hand, they didn't have anything else available to eat. The tame ones I shoot are really wild, out in a county park with very few people around. Some stop by to feed them, but not like a full feeder all the time. So when I have food, and theres no other stuff to eat, they get brave and come to my hand.

A way to get close to them, is to put a lawn chair out there and let them get use to it for a day or so. Then just go out with the camera and sit in it. I do this here when I have the time. It's cold, so I put on the coveralls and go out there and wait for them. My neighbors think I'm nuts. Eventually, the birds and the neighbors will just ignore you if your not moving around. But, the best way to photograph them is to get a DSLR and a long lens. And you'll still have to get close to get great shots.