For research, I tend to focus on the tier 2 buildings and such first, so I could build up a foundation for my empire (now it's output bonuses), then I would do the same for ship tech, then I would research as needed after that.
One way to at least help with his issue, is get bonuses that increase research options, 'cause that seems to be one of the problems. There's a tech to add +1 research options, discovery adds +1 research options, and technocracy adds +1 research options.
If you have a few labs, the research speed/tech pool should be high enough you could be able to roll the tech you need. However, since you want specific techs, I'd look into each kind of tech you want, and try to hire the appropriate scientists to increase the chance of rolling specific techs when you can. In the case of military techs; Ships and starbases need Voidcraft, combat computers need computing, weapons and reactors need particles, shields need field manipulation, kinetics, engines, and missiles need propulsion, armor needs materials, and so on. You could have these scientists on science ships so you have a use for them, and they can level up while exploring. Then you can switch them out when needed, keeping them on research for a few techs to see if you can roll the one you want.
As for vassal's, I don't know, sorry. I don't use those often.
