1) You really haven't played it much. I suggest trying a set of new strings,tuning it to a tuning it will hold and you are familiar with playing, and STEP AWAY FROM THE TUNER ! Play it for a few days, and see what your ears think. If you ask a scientist what time it is, you may get an answer such as "7:42 and 13 seconds, P.M., Eastern Daylight time." A poet may tell you "It's that time of day when the trees look like black lace against a fading sky." You have asked the scientist (the tuner), how well it plays in tune. Now pose your question to the poet (your ears.) Who do you wish to please when you play your dulcimer? The tuner, or your ears? If it plays close enough that your ears are pleased, you probably don't need to do anything else but enjoy your new dulcimer. let the tuner do it's necessary job, but only ask it scientific questions. It doesn't know Robert Frost from Helen Steiner Rice. NOTE: I suspect many flesh and blood scientists also know and understand poetry, but none of my electronic tuners have ever exhibited any similar knowledge.
2) A deceased luthier whose work I admired and whose word I trusted told me frets should be within .001" to be truly accurate. He may have been more critical than most, but 1/16" is probably not close enough by any standards. If you measure each fret from the nut/zero fret, they will be more accurate. If you measure each fret from the fret before it, any error gets added to each fret after it. Several small errors can then be compounded into large errors up the fret board. Not knowing who built this dulcimer, we can't ask how, or how accurately, they were measured.
3) Old strings can be hard to tune, and may not play in tune, so a new set may be a good start. If the bridge is movable, carefully positioning it is the next step. If the bridge is not movable, small changes can be made by compensating the saddle. This may be best left to someone with experience, but if the saddle is easily removed from the bridge, a new one can be made at a very small cost if you mess up, so you may want to try compensating it. First make sure it can be easily removed without damaging the fret board or top of the dulcimer. I'll leave compensation adjustments to someone better equipped to explain. I understand the concept, and have seen where it is used on some of my instruments, but have very little experience actually doing the adjustments.
Paul