First of all, make sure you're feeding Cosmo a high-quality food without lots of fillers. They go through the fillers quickly and are less satisfied, so they start looking for food. Secondly, I think it's very common for kitties who are not used to having regular meals to forage for food. It's even common for kitties who are used to regular meals!
Once he gets older and gets settled in your home, used to the feeding routine, he'll be less likely to forage.
Also, he's just a kitty and kitties are quite curious. They like to get into EVERYTHING and explore everything! So, being on the counters is just him getting to know his environment and becoming familiar with his surroundings and feeling at home. However, now's the time to start training him to stay off the counters if you don't want him there. We had to do this with Hannah, and it took constant attention to train her, but she did eventually learn that she can have most other high places in the house except the kitchen counters and the tables. Get your kitty a cat tree so he'll have his own high place to be. Put it near a window, so Cosmo can see to his outside world and watch the birds and such.
We saved soda cans, cleaned them out, then after they dried, filled them about 1/4 to 1/2 the way full with dried pinto beans. We taped the tops closed with duct tape. Then, we lined them up along the edge of the counter, back a bit so she'd think there was landing room. When she'd jump, she'd land near the cans and they'd rattle and scare her. We also keep a few cans handy throughout the house. When we'd catch her terrorizing our other cat or getting into something she shouldn't be in, we'd rattle the can and sternly tell her "NO!". We didn't yell, but very firmly stated NO. Also, we removed her and told her "NO!" Normally that was followed with "That's not a kitty spot." (It still is said frequently in our house and she's been here 3 years!)
You can also try putting foil on the counters. Supposedly kitties don't like the sound of foil, but foil didn't stop Hannah. As a last resort, you can go to Lowe's or someplace like that and get plastic carpet runners. They have little nubs on the bottom to hold them in place on the bottom. Cut them to the size you need and turn them upside down on the counter (or other surface where you don't want kitty). The little nubs won't hurt their paws, but they won't feel comfortable either. They'll quickly learn not to be in that spot.
I'm not a fan of the squirt bottle, but have had much more success with the above techniques than any squirt bottle.