The Sins of a Solar Empire Trinity in stores comes with only Sins Diplomacy 1.0 or 1.01 (don't remember which), you will need to download impulse after you get it to upgrade to 1.2.

Since you already have the base game, it would be better to buy the Entrenchment and Diplomacy expansions digitally from Impulse ($20 instead of $40). This is the exact same as getting Trinity, except that you save $20. They add a lot of cool stuff so its probably worth it.

Then do so. Just don't attempt to hook up both to IronClad Online. That would be a big no no.

You just install Impulse on both computers and it will install the expansions on both. There aren't any download limits or anything (at least that a normal person could reach. If you put it on a thousands of computers SD might get suspicious). 

Just make sure you register your GotY game if you haven't already, then use that same account to purchase the expansions and you'll be good to go (the only time people tend to have problems is if they have the games under different account names).