- How do I best resist Morgoth?
Are we actively resisting him, or just waiting out the storm? I'd have tried to catch Manwë's ear and tell him that Melkor thinks that he's full of hot air. And that Varda likes decisive valar. Anyway, unless you could clone Fingolfin and launch the clones into the fray right as the original elf starts to falter, your best bet is to ride out the years until the War of Wrath.
- How friendly should I be with Thingol?
Very. See if Melian has a sister. But don't sign any mutual assurance treaties. See if he would like to vacation at your place (see below), as long as he brings the wife.
- How far should I co-operate with my valiant, elficidal jewel-seeking relatives?
Be polite, but say that you are preparing your own attack forces and so aren't able to send help in the current battle. And leave your mailing address as "Gondolin."
- What sort of site would be good for my kingdom?
Isle of Balar. Think about it. An island, ringed by water that never was friendly to Morgoth, protected by a Melian-type girdle. Also, I'd build an eyrie just in case some eagles wanted to take up residence.
- What do I do about Dwarves and Men?
Treat them as you would the other elves. Trade, but don't let them inside.
- What sort of culture am I ideally aiming for?
A gathering of seafaring independent people that are ready to split south at first sight of orcs/Balrogs in boats or when the Girdle starting to fizzle. Each subtribe would be able to restart the colony as needed wherever they found safety should the worst befall the base. Faced with the indomitable Morgoth, it'd be a good idea not to be too visible, or too stuck in one place, or too hung up with petty squabbles.