Hey blackjack, going through your questions below one by one!
-There's no built in functionality in the game for taking screenshots and I'm not sure if itch has one. Windows has a screenshot function though if you press Windows Key and Print Screen. It will save to pictures/screenshots.
-Health pickups drop from enemies you kill and from crates. It sounds like you're just having trouble making it past the first few enemy encounters though. To improve your combat I'd suggest practicing using the dodge-roll and melee to maneuver, stun, and finish off enemies. The combat becomes a balance of dodging enemy attacks and replenishing health from killing more enemies.
-There are passive mods that will improve your pistol/weapons, but if you're having trouble finding new weapons in the first place you should look for stockpiles. The yellow ammunition waypoints on the map (M) always have a guaranteed weapon spawn, but you'll need to venture out of the intro area to find them usually.
-DNA can drop off of any enemy in the game. If you haven't been getting very far it's likely you just haven't fought enough enemies to get a piece to drop. In the next patch DNA caches are being added so if one spawns you'll have a chance to get a boost of DNA without needing to get lucky fighting enemies. Once you die you can use any DNA looted and each piece has its own chance to mutate (with about 60 mutations in the game currently and more coming soon).
Let us know if you have any more issues or questions!