The Diana 350... In my opinion, probably one of the best break barrel platforms. At factory power expect 18-21fpe for the mighty 4.5mm, I'd say the majority of newer models will be in the 20-21fpe range, at least my two T06 examples in .177 were. If you have a model with a globe sight, which is wonderful, you could get a Truglo globe sight very cheap when on sale (check Fieldsupply website now & then). While the Truglo isn't built as solid it makes up for it by being very visible during certain lighting conditions. Also, the Truglo sits lower, so with your rear sight about bottomed out you'll be zero'd about 30 yards. With just a few clicks you can be shooting targets out to 80 yards open sights, or further provided you have room and can see the target.
If keeping at full power, expect the OEM piston seal to last about 3000 rounds, but this isn't a "hard" number. Some seals might last twice as long. Just saying at about 3000 pellets in, I'd recommend getting a replacement seal or two to keep on hand. No real need to get a tune kit (like PG4 or whatever), but they are available. I do like Vortek springs, they fit the OEM guides very well. If you want a mild shooter, drop in a Diana 48 spring. This should knock the power down to about 15fpe, then you can shoot "normal" weight pellets. If you decide on a custom spring for even more power, I don't recommend anything with a wire diameter beyond .135" or you risk cracking your stock (if over sprung w/too many coils). I've hit 23FPE with the 350 platform in .177 without really trying, but the gun wasn't happy. Either way, the Diana 350 is extremely easy to work on.
The rifling Diana uses is a bit different. They use wide lands with narrower grooves, but it works great for pellets. My reduced power (~15fpe) 350 can accurately sling wadcutters out to 40 yards on calm days. Great for plinking or pesting. If the wind picks up or if I need to shoot further, I'll switch to H&N FTT's. That being said, my 350 barrels aren't really pellet picky but don't expect Winnie Domes to hold groups past 30-35 yards, hehe. My high power 350 currently has a PG3 kit, I believe. Power was lower than expected, but I spaced it up to make 18fpe. There wasn't really twang with the oem internals, but power was a little higher than I prefer so I tried the kit on a whim. It works well but not totally necessary. Most pellets in the 10.5gr range work very well and stay subsonic. My preference for those heavier options are the H&N Barracuda Hunters (not the extremes as they're too light).
As for what pests/game animals can be taken with a full power Diana 350... It depends on local jurisdictions, personal ethics and what pellet being used. Generally, it's not recommended to take anything larger than a raccoon with any break barrel, and some may argue raccoons are beyond the limits... while others... in certain circumstances... have cleanly taken larger critters (edited to avoid conflicts) ... "allegedly"... But that's a whole different can of worms I don't care to open...
But I sure do like these emoji friends...
Also, keep in mind the 350 has a lot of swept volume, it's essentially a 34 long (as in a D34 with elongated compression tube). There's a bit of a snap to the muzzle report. If you have the model with a globe sight and wish to minimize that snap, search out a Weihrauch 16mm slip on moderator with cut out for globe sights (AOA has them occasionally). Otherwise, any 16mm unit or adapter should work if you wish and are willing to give up open sights.
That's about all I've got. Enjoy your new to you gun.