What kind of trade goods from dwarven caravans can I expect if I become a capital? Will they send me steel (or at least steel goods)? How about gold, silver, etc.? Silk, clay? I assume the wont send me stone, except as blocks.
Only one of my forts ever became a Mountainhome with a King, and since then I've avoided this honor like the plague: the dwarven caravan dropped to near-elven levels of (lack of) goods. The other two caravans were unaffected, but the dwarven one dropped to only 2 wagons from 5-6, and started bringing utterly useless goods.
Guessing in the dark, I presume your issue is that you would like the caravan to bring certain items? Because that's what trade agreements are for. You can order bars and stones, with your chosen level of priority, and the next caravan will bring some... at an overprice, which starts at 110-130%, for priority 1, and goes up to 200-220% for topmost priority.
Some things to keep in mind when ordering goods:
the level of priority seems to translate directly into the number of brought items: order silver bars with priority 2? You get 2 silver bars. Especially with bars, you might be better off gambling on standard caravan autogeneration (or setting traps and mining goblinite).
It's pretty good to use on stone, which you'd normally not get (except clay boulders), and it absolutely shines on leather and cloth: these are counted by the bin, not by the item; ordering every type of leather at priority 1 will result in merchants bringing a bin of 10 hides for each type, which translates into > 250 tanned hides, at relatively small overpricing - a skin which used to cost 6☼ now costs 7☼, big deal.
overpricing stacks. If you order, say, both iron bars and iron anvils, the 3 bars used to make the anvil get overpriced, and then the anvil itself gets overpriced. I made the mistake of ordering both, and ended up with a ~250% price hike, for priority 2 bars and priority 1 anvil.
Having a monarch and/or being the capital currently has no effect on trade.
The goods available to your civilization are determined at World Gen, and (as of now), those materials do not change over the course of play.