1.) How do I declare war without using an attack/invasion vessle?
AFAIK in DL you can't. But you can attack something very "cheap", or you can use "passive" declaration: just park some of your troop transports close to AI's planets any THEY will declare war on you eventuelly.
I like to play as realistically as possible, and I don't like to surprise attack people if I can help it.
In reality the most of best attacks were the surprise ones.
2.) I read somewhere that the computer doesn't cheat. If so, how does it do so great (research, production, etc) without building as much farms/econ buildings as me ...
At "though" level AI doesn't "cheat" (it gets exactly what you'd get). But it uses extreme slider settings a lot (very high taxes, very high production of ships...) when it calculates it would be beneficial. If you want to compete with them, you too must fiddle with sliders, at least for each particular phase of the game (e.g. whan you realize you'll really neeed some tech, you set research very high, until you get it; or when you're expanding, you set your mil slider very high, and set focus on social production on non-production planets).
...when I check through espionage?
Please be advised the info from a spy on a planet (DA feature) shows the MAX production a planet can pump out, not the actual one. Only when you get the second level of espionage you see actual production (in DL it's IIRC when you spend 500 BCs in espionage on that race).
3.) When fighting a winning war, my econony often goes into shambles because of the large influx of empty planets
The usual way to decrease costs of newly conquered planets is to dump there 1-2B pop, so that these planets start producing some taxes and with them cover the maintenance costs of buildings and production. Yet another way is to scrap some (usually the most expensive) buildings on that planet, or change (upgrade) them into econ buildings. The last and the most radical way is to destoy the entire colony. Beware though abandoning a colony will lower planet's class by 2-3 points in DL and set it to 0 (uninhabitable) in DA.