Before messing with any bar height on that bike, make certain you have the spacers underneath to remove. If not, then don't even attempt. And don't be in a hurry to cut your steerer tube. Just put the spacers on top of the stem. You can cut the steerer tube when you are certain of where you want your bar height. Or... don't cut it at all. I don't know what the bike had that you rode previously, but with a threadless headset, when you change the stem height, you'll also be changing how loose or tight the headset bearings are. Easy when you are used to it, but can be an issue the first time or two. Especially if you aren't aware.
Looks like next natural step after fixing the position - what type of tyres would you recommend for riding in a parks and bike paths?
Probably, but Specialized makes a decent tire, And if those are the
Nimbus II Sport Reflect they are probably good enough to ride till they wear out. And they also look reasonable for paved trails and streets. As I said, I don't think you'll find the ride difference between tires to have much of a difference for the issue you describe. Perhaps on long rides it may.
Typically, I think the more tpi in the casing, the more supple the tire. That's a good thing for both rolling resistance and comfort. So when it's time to change tires, perhaps consider that. However in the time it takes you to wear them out, there'll be more new models out there and some of our current favorites will be gone. Tire makers are changing up things frequently and even a slight change in the full name of a model might mean an entirely different tire in terms of ride and performance.