[Lalaith]

Hmmm....first off, my initial problem was that the flight of the Noldor from Valinor was an essentially quixotic venture, about as unMachiavellian as it could possibly be. Sure, the various family branches came with different motives, but none of them could be described as pragmatic. In other words, Saucie's analysis is pretty accurate.
The second problem is that of contributing to history. For survival tactics based on neutrality, the obvious modern-day blueprint to follow is Switzerland. But how interesting is a history of Switzerland? William Tell, Zwingli, banks and chocolate are the high points, not enough to fill more than a side note in a great heroic epic...
Anyway,
How do I best resist Morgoth?
Initiate a well-funded and comprehensive programme of training and education. "Orc Control - the Neuro-Linguistic Route." "The Seven Habits of Effective Warriors". "The Lies of Morgoth and how to nail them." "Silmaril 14-Day Detox". Operate a policy of affirmative action for your courses. Make sure you have a good ratio of non-domestic students, and that you fill your quotas of non-Edain mortals, dwarves(however petty), the Avari. These were a much-neglected pool of Morgoth-fighting potential.

How friendly should I be with Thingol?

All overtures should be made through his much more sensible wife. Perhaps she might be interested in funding a Melian Scholarship?

How far should I co-operate with my valiant, elficidal jewel-seeking relatives?
Get them on the Sil detox programme, of course.

What sort of site would be good for my kingdom?

Do I go for:
A The cave network approach
B The hidden city approach
C The enchanted forest approach
D The seaside haven approach
E The massive great try-this-for-size-Morgoth-old-chap fortress approach

Hmm...my instincts veer towards C or more likely D (Eglarest, anyone?) The educational programme is of course a front for the surreptitious shipbuilding. This has a double benefit: your people have a means of escape if attacked, (Morgoth didn't seem to have any Corsair-style naval fleet at this point, did he?) plus the long-term goal getting the Valar eventually onside seems to combine the quixotic and constructive in a rather satisfactory way.

What do I do about Dwarves and Men?
See first point.

What sort of culture am I ideally aiming for?

I can't better Feanor's proposal - all except the Marmite. Bleugh. Get the Dorwinian trade going with some imports of champers, instead.