"The tuners, as others have mentioned, are terrible. Really really bad tuners. They seem to have a lot of 'give' before they actually turn the string end. I'm not sure whats going on there, but they are hard to use"
Actually, the Schoenhut's tuners are a decent version of a Grover. All friction tuners need adjustment
before attempting to tune. This is actually pretty easy - one common method is to completely loosen the strings, then slightly tighten the adjusting screw until you feel a little friction. That should do it.
Since excellent intonation and easy fretting are especially important for beginners, it's hard to recommend a Dolphin, whose setup will be of the quick 'n dirty variety. It will be playable, but both the action and intonation cannot be predicted, and at 2 to 3x the price.
A word about the bridge: even mine has a visible but inconsequental "gap" between the bridge and the soundboard, as it is not glued on, but rather attached with screws from below. Not an issue.
BTW, it appears your tuners were slipping, which is not a fault of the tuner, but does mean that you likely need to tighten them. You can proceed as above, or you can loosening them a turn, tightening the screw a quarter-turn, then tighten again, and repeat as necessary. The adjustment screws are there for a reason.
When I did a search for comments on the Schoenhut, there were a number of posters who had loose tuners (which is the proper way to ship the instrument), but who had no problem in adjusting them. Once adjusted, they are extremely reliable. With my Martin 600 strings our Schoenhut holds tune better than any of our four other Kala's with Aquilas and standard tuners.
And string selection: do not - I repeat - do not put Aquilas on this. Enough of us have used and strongly recommend Martins for this uke. The reason is that due to the plastic design, the Schoenhut is already quite bright, and will be overpowered by the Aquilas - loud, harsh and dull. The Martins provide much improved clarity and good sustain, almost harplike, lovely - and they are easier to play. But it's your money...