I searced for information on refineries but the search function of the forums only shows up as "no response to your search"...regardless of what I search for. am I missing something on getting this to work?
The forum search isn't all that great. I'd recommend using google and appending 'site:forums.sinsofasolarempire.com' to the search, without the quotes obviously 
How can I put a refinery in trade mode, so that it makes cash instead of resources? I've tried with trade posts, with refineries, and with extractors, but dont see the option.
Generally, you cannot. For the TEC and Vasari, these are separate structures. The Advent only have a trade port structure, but you can research an ability for it that switches it to boosting resource production on that planet.
Is there a way to quickly tell how many military labs/science labs that I have? And do you usually build them a couple at each planet until you get what you need, or one at a planet as you expand?
I think hovering over the research button on top shows you your lab numbers now? It might be an Entrenchment thing, but I think it's also in 1.1. Alternatively, the research screens themselves shade based on how many labs you have. It will be greyed out for the tiers with no labs, so if 3 tiers are greyed out you have 5 labs.
and how quickly do trade ships begin producing from a Trade post? Is there a limit, or is this gauged by the number of other trade posts under your control (one ship per?)
I believe it's 6 ships per trade port? They start producing immediately, but it takes a few minutes to pump them all out.