There are kevlar tires that resist punctures pretty well but the best thing you can do to prevent flats is to watch where you are going and to always make sure the tires are properly inflated.
Your local bike shop can tell you if it's worth fixing up your old bike or not. I imagine some maintenance and a tune-up will do wonders.
As for the other two bikes, they are on the low end and will likely need a bit more attention paid to them regarding adjustments to keep them running smooth.
Since you live in a hilly area, you should pay attention to the gearing of any bike you might choose to purchase. A triple set of rings in the front will give you more low gears for those hills. Also, for the rear gears, the bigger the number, the lower the gear. (eg. a mountain bike cassette could be a 12-34, meaning it has a 12 tooth cog and goes up to a 34 tooth cog. Most road bike gears in the back are 12-25) For the front rings, the lower the number, the lower the gearing. A triple crankset may typically be 53-39-30 or 48-36-26; depending upon the bike.
Hybrids typically have road bike wheels with fatter tires, a front suspension fork, upright geometry, and maybe a suspension seatpost and/or an adjustable stem to move the handlebars front or back. Generally, you have to dismount the saddle to have your feet touch the ground. IMHO, the suspension fork isn't needed unless you are jumping off curbs. The ride is comfortable, you can pull a trailer. The gearing is usually wide enough to climb anything while still being fast enough on the flats.
Comfort bikes are like the K2 Big Easy Deuce. The saddle has you way back on the back wheel, the pedals are forward, the geometry is upright, and you can put your feet on the ground without leaving the saddle. Slower than a hybrid, the ride is comfortable, especially if you have back problems. Again, they usually have that worthless front suspension which just adds weight and robs power but you aren't riding a comfort bike in a race.
The K2 is a comfort bike. The ride will be comfortable and smooth, thanks to the fat tires and front suspension and upright geometry. It'll be slow, too.(I'm comparing the ride to a typicaly road bike) Gearing is low enough to climb walls but the geometry is less efficient but you'll still be able to pull that trailer. Components are on the very low end; I expect you'll need to be vigilant in keeping things adjusted properly to keep the bike running smoothly. I'm guessing you ride recreationally a few miles at a time so you should be okay though I'd save a few more dollars and get something better from Giant or Trek.
The Schwinn Traveler is very similar in design to the Giant FCR or the Trek FX but the components are very low budget. It should be faster than the K2 due to its larger wheels, thinner tires, and more aggressive geometry.
If you do get one of the bikes you mentioned, please take them to a bike shop to make sure everything has been assembled correctly and is properly tuned.
The Trek Pure is a comfort bike that is fun to ride and should be in line with what you are looking for.
7.2FX is similar in style to the Schwinn but a much nicer(and more expensive) bicycle
My wife had a hybrid but she outgrew it; it was too slow for her. She loves her 7.5FX
Check with your bike shop about tuning up the old bike first. If it turns out to be unsalvageable; please consider spending a bit more on a new bike. You'll love the ride so much better, you will ride more and farther.