@Phroedrick, this is a hurdle you will conquer and it will cease to be a problem.
It is rare that the nut is not secured, but a floating bridge is pretty common. When that is the case, you will want to change one string at a time so that the bridge and/or nut basically stay in position while you put the new strings on. You can always adjust positioning later if the intonation is off, but changing one string at a time will make things easier.
The order does not matter.
The direction the tuners tighten the strings depends on which side of the post the strings are strung on. If the posts are horizontal, you want the strings to go over the top of the post. If the posts are vertical, you will want the strings to go on the inside. The fact is that it will work either way, but those are the standard conventions.
One trick to keeping the string in place is to use a capo to hold it down. Put the loop over the brad nail or whatever is there to hold the end of the string, pull it toward the head of the dulcimer, and then put a capo on to hold the string in place while you wind it around the tuner. I learned that capo trick from @Butch-Ross and am embarrassed that I hadn't thought of it myself.
Here is a video of @Guy-Babusek demonstrating how to string a dulcimer, and here is a video by @Bing-Futch doing the same.
EDIT: @Salt-Springs beat me to posting those same two videos. Great minds . . .
updated by @dusty: 07/12/19 09:24:18PM