You have a very good question about the tuning issue from side-to-side across the treble bridge, Mousetrapezoid. This happens to almost every player eventually, but usually you will not have to move the bridge to fix it unless it is really, really, really out of tune. I always advise newish players not to move the bridge because it is way too easy to truly mess up everything and make the whole dulcimer out of tune.
Here are a few questions to ask yourself before you move the bridge:
How bad are the notes out of tune from side to side? Is it a tiny bit out? Is it a half step? Etc. Judge how bad it is.
Does 1 of the 2 strings tune correctly or are both out of tune?
Is this problem only at the top or is it the whole length of the treble bridge?
Do you have metal/brass rods that the string crosses or is is delrin? What are the rods that are on the sides....at the tuning pins?
Before you start moving the bridge, I would try the following things first:
Before you tune the strings, push down hard with your fingers on the set of strings that won't tune on either side of the bridge. Push down on one side then the other then back and forth a few times. They may need to be stretched a little or they may be caught on the bridge where they cross. Now try to tune starting with the high side and see if the low side comes in tune.
Sometimes if you just start with the high side and tune that set of strings first, the lower set will come in tune. I find that to happen often with my tuning.
If that does not work, loosen the strings about a half turn. Take your fingers and put them under the strings near the bridge and lift them off the bridge. While you have them lifted you can look at the bridge and see if there are any unusual grooves where the strings sit. If there are large grooves, you can move the strings just to the top of the old groove or just to the bottom of the old groove and retune the strings. You will probably have to hold them in their new place when you put the tension back on. They will want to go back into the old grooves.
Also while you have the strings loosened you can check to see if there are grooves in the side bridges. You can move the strings out of the groove and hold them in place while you tighten them again.
Often you may find that there are a couple of strings that if you tune the higher set, cross the bridge and tune the lower set, they will both go in tune no problem. This sometime happens to me. I just touch the tuner wrench to barely move it and pluck the string back and forth across until both sides are in tune.
If all else fails, move the bridge or better yet, have someone who has some experience to move the bridge. The reason I hesitate to tell you to move the bridge is that it is really easy to move it too far and then hard to get it back because the move has to be so small in most cases. And if it moved too much, the bottom part of the bridge may move and then you will have the bottom bridge to put back in place. It is just really tricky. I know you can do it, it's just that you need to know how easily it is to get it out of whack! (
I would be glad to look at it and fix it if you can make it to the DE workshop coming up this month!