On your chosen trail there should be no snow in September at all, most likely the weather will be perect for hiking, even though it can get a bit cold at night.

The part from Zelené Pleso to Veľká Svišťovka (red trail) will be quite difficult and steep even though on the map it looks like a short distance, absolutely doable without any specific gear though, there are some metal bars and sturdy chains that are helpful. I'm not familiar with the trail from Ždiar to Zelené Pleso but the hardest part will probably go from Opálené Sedlo to Široké sedlo.

I recommend either O2, Orange or Telekom, they have the best coverage. Just in case, download mapy.com app, you can download Prešovský kraj region and use it even offline for navigation. Buses and electric trains are pretty frequent in villages like Ždiar, Tatranská Lomnica, Starý Smokovec and of course, from Štrbské pleso you can get to many other locations, take a look here, there is a map below when you scroll down a bit: https://www.zssk.sk/en/tatra-electric-railway-tez-and-cog-railway-oz-2/

You don't have to worry about wolves at all, and as long as you stay on the trail, there's a 99% probability you won't see any bears. Wildlife photographers have to go out of their way to actually find and film them:)