Wood pellets in a smoke box:

First off, I'd cease using paper as your tinder. While it is the cheapest way to get your charcoal started, it does create a lot of smoke and ash flying around. And avoid oil-soaked paper. This will create even more smoke (I speak from experience).

I started using firestarter cubes, as they burn hotter and longer, meaning my coals are ready for cooking faster. Also, another side effect has been that the only smoke I see is from the charcoal itself. I started using the Weber ones, but they are expensive and don't burn as long. So I switched to a different fire lighter. You'll have to forgive me, as the name escapes me -- once I get home I'll edit this answer with a picture of the bag. Anyway, they're cheap, at about US$ 5-6 for a bag of 72. And they burn long, hot, and smoke-free.

The only remaining problem you'll have is that when charcoal begins igniting from a low starting temperature, it puts out quite a bit of smoke. If you are still worried about smoking your neighbors out, you will probably want to experiment with different brands and types. Stubbs briquettes are a bit less smoky on ignition, as is most lump charcoal. If you use lump, though, you should make sure you pack the charcoal ring in your WSM nice and tight, or it will burn very hot and fast. Also, put a piece of expanded steel on the charcoal grate to keep the small pieces of charcoal from falling through.

You can use a gas grill for smoking and indirect cooking. Run the burner on one side of the grill, and place the food on the other. Wrap wood chips in foil, poke a few holes in it, and place the foil pack over the active burner. You can soak the chips if you like. I've never seen much point in that, myself. The water evaporates rather quickly, and you're back with dry chips. Better to limit the oxygen supply to the wood so it will smolder, and place the smoke pack somewhere so that it does not have too much heat applied to it.

It absolutely does matter what kind of wood you use. Avoid soft and sap-laden woods like pine. Stick with hardwoods or fruit woods. Hickory is often paired with pork. Apple and cherry are used ubiquitously. Oak is strong, but imparts an interesting flavor. Mesquite can be too powerful for some people -- I personally use it for grilling, but never for smoking.

Finally, be aware that you are going to have some limitations with your gas grill. It neither holds nor convects heat in the same manner that purpose-built smokers and some charcoal grills do. You are more likely to have hot and cold spots on your grill, so you will need to examine what you are cooking each hour and adjust accordingly if necessary. You will also not get nearly as much smoke to your food, so you will also have to adjust how much wood you use accordingly (though many people make the mistake of too much smoke in their barbecue, so a gas grill's limitation in this regard can be a benefit).