1. Is there a reason for me to record to my Sony PCM-M10 and then dump it all to my computer later, or should I just run a line out from the PA in to the computer and record direct to my hard drive? (My computer will handle the task just fine.)
If you want to up your recording game, the space you are in is going to matter more than any other aspect of your setup. The finest mic driving the finest pre-amps on earth, when stuck into a crappy space will give you a super delux high fidelity recording of a crappy space. So treat your room. Get a copy of RoomEQWizard or something similar and measure the heck out of wherever it is you're going to be making recordings. Then treat for what you find. That alone will make a noticeable difference.
Once that's done, then you can look at kit. And as far as is there a reason to not use the Sony... yes. Two I can think of right off the top of my head.
1. Tracks. It's a fine field recorder, but the best results for recording a guy on an acoustic guitar is to close mic the guitar (in stereo if you can) and close mic the vocals. And if you have a nice space, mic the room with an LDC. Then you mix that down to a nice representative stereo track. The Sony gives you a coincidental pair of mics, which is great for a broad stereo image (think drum overheads or choral groups), but not so great if I need to mic defensively or close mic specifically. So if you're sorta serious about better recordings, you are going to be limited in what you can do with it when you mix.
2. Mic options. As I said, the Sony has a pair of SDCs in a coincidental pair configuration. That's great, but also limited. Vocals often sound better when recorded with an LDC or ribbon mic. Same with a room mic for ambiance. The SDC are fantastic for close micing the guitar, and while you can do that with a coincident pair, I prefer to mic the neck and the tone hole or resonator board individually. Using an external interface allows you flexibility with mic selection that you won't get with the Sony.
Good audio interfaces are cheap, and readily available. Focusrite, Behringer, Tascam, Zoom... They all do the job at an affordable price point.
2. Do you all use any effects on your instruments or record using a dry signal and add effects later?
My general philosophy is to record the sound I want, and I'm kind of rude about it when it comes to amp tones. If you want "LP through a Marshal" tone, then bring an LP and a Marshal, and we'll record that. Don't bring me your ES-335 and your Fender Delux and expect to sound like Slash because magic plugin X says you will. I'm less finicky about it when it comes to things like EQ and compression.
I've also never met a digital reverb I liked. The ones that have come closest to tolerable have all been convolution reverbs. I've essentially given up on algorithmic verbs altogether.
3. With that cheapo Dean Markley acoustic pup that mounts in my sound hole, will I need some kind of preamp to boost my guitar's signal or does that sucker drive adequately?
Drive what adequately? That depends on your pre-amp. Most modern interfaces will let you switch between mic/instrument levels, and line levels. If you're concerned, spend $20 on one of those ART single tube pre-amps. I've got two of them. They're great.
Start with the room. Seriously.