Does it matter if your demense is connected for techs to spread from your provinces. For example, I'm wondering if I should give up my Capital and move to the Coast or if it doesn't really matter? My current capital is inland.
It helps to keep your demense together.
Among the bonuses to tech spread are a demense bonus (25% per holding in your demense with the next complete tech level) and a neighbor bonus (30% per neighboring county with the next complete tech level).
Normally, a smart player is going to be a tech leader. He's going to use his advisers to increase tech in his capital, he's going to build university buildings and rush town infra to improve them, and he's going to have much better than average kings and advisers to increase tech rate. So counties in your demense are going to be much higher tech than neighbors and are going to increase faster.
What this means is that when you split up your demense, counties hit a "wall" of slow tech when somebody else borders you. Your neighbor is not going to get your massive demense bonuses. He's not going to build universities. His advisers are going to be mediocre. Your high tech counties next to his help with the neighbor bonus, but they increase tech much much slower than they would if you held them with all your additional bonuses.
Splitting up your demense therefore puts massive tech slowdown zones between your counties. It effectively depirves you of the substantial neighbor bonuses because your neighbors are laggards. So you'll increase tech slower overall.
You can decide how much this matters to you. It's not the worst thing to build two compact duchies of high tech separated from each other -- although the one without your capital will always advance slower than the one with it. But splitting your demense up all over the place is tech suicide.
On the other hand as a king it can help to spread out as that way your supertech can give a +30% bonus to tech to more of your vassals.
That last sentence is my strategy so far, pick a one or two duchy core, add Sicily, or an Iberian or Levantine duchy, then let the high tech spread to my homeland.It can make sense to "seed" a few personal demense counties in low tech parts of your empire. That way you can make make one county in the area catch up, giving a neighbor bonus to a lagging area. But you do that at a cost of hurting your own personal tech. Because you're going to have one less demense bonus / neighbor bonus county working on your base area.
So you can stick a few around once you're a clear tech leader. But it's a bad strategy to do more than that. It worth far more to trade that lagging county for one in a high tech area to then drag high tech back to your heartland.