King can never be a vassal of another king, and there is no “partial vassal” or something like that (and the same applies to all the other levels too).
The duchy will become part of your kingdom (de facto, this won't change which kingdom it is part of de jure). But if the crown authority in the other kingdom is high enough, you won't get the duchy (it prevents titles to pass out of the realm through inheritance).
You may have read tips relating to an earlier version of the game.
Now, if I recall correctly, technology advancements were more or less random, although you could set a focus area for each group (e.g. "light infantry"). This has been changed, and you now have more direct control over the technology.
Similar to Europa Universalis, you gain technology points over time in each of the three groups. The amount of points gained, depends on your ruler, your council, ahead/behind bonuses etc.
When you have enough points to spend, you click on the "advance technology" button – it's the button with the magnifying glass and arrow in the image below. Note, that technology doesn't cost you gold – just arbitrary technology points, which have no other use in the game.
Now, just because you have points to spend, doesn't mean you should. Since every level increases in cost, it would often make sense to save up for a more expensive advancement.
Also, some technologies will slowly improve on its own. For example, you might get a passive increase to "Church Infrastructure" by building a bunch of religious buildings. You can see that by hovering over the cogs, which should be spinning and have a slightly different colour (see screen shot).
Image borrowed from http://www.ckiiwiki.com/Technology
Best Answer
If your capital moves to Middlesex, you will use Middlesex's technology levels.
However, you will keep any technology points you haven't spent, and can spend them in Middlesex after the move. If you spend some time saving them, you'll be able to give Middlesex a technology boost when you arrive.