tHE Connections Devices and 90S Devices Can connect to Mix.
No... they can't? The 90s devices can't even connect at all.
OP, I'll answer each one of your requirements individually to give you an idea of the range of Tamagotchis there are:
-Has a decent amount of pets
The originals only have about 11 characters so I'm guessing you want quite a bit more than that. This condition eliminates pretty much all of the vintage models, except for the Osutchi and Mesutchi, which have quite a few characters. Of the modern releases, here's the ones that have the most characters - though be aware that some of these characters might be inaccessible due to requiring external devices or web services that no longer exist:
International releases:
V1: 20 characters
V2: 51 characters
V3: 50 characters
V4: 52 characters
V4.5: 52 characters
V5: 65 characters
V5.5: 60 characters
Music Star: 40 characters
Tama-Go: 36 characters
Friends: 36 characters
Friends Dream Town: 36 characters
Japanese-exclusive releases:
Keitai: 23 characters
Hanerutchi 1: 21 characters
Akai: 23 characters
Entama: 56 characters
Uratama: 58 characters
Hanerutchi 2: 46 characters
Oden-Kun: 22 characters
Plus Color: 26 characters
Hexagontchi: 26 characters
iD: 21 characters
iDL: 48 characters
P's: 42 characters with an additional 54 characters unlocked with pierces (sold separately), I'm not sure if all the additional characters can be raised though
4U: 16 characters and 16 personality stage forms (not sure what these are but they're functionally different characters, I believe) with an additional 32 characters unlocked with downloads and touch cards (sold separately), again I'm not sure if all the additional characters can be raised
4U+: 18 characters and 22 personality stage forms, as well as (I assume) the 32 downloadable characters from 4U
M!x: There are 5 different versions of the M!x and they each have 22 characters, plus the ability to "mix" characters together through breeding to get a seemingly endless number of variations
Meets: 19 characters as well as the aforementioned mixing mechanic
-Can connect with other Tamagotchi devices
All the Tamagotchis mentioned above can connect to others, as far as I'm aware! There are restrictions though:
-Osutchi and Mesutchi can only connect with eachother
-The V1, 2, 3, 4 and 4.5 can all connect, though Tamagotchis of different versions connect as though they're V1s which limits the number of things to do
-The Japanese version of the V1 - the Plus - can connect with the Keitai and Akai
-The V2, 3, 4 and 4.5 can all connect to the Keitai and Akai too
-The Keitai, Akai, Entama and Uratama can all connect to eachother
-The V5, 5.5, Music Star and Tama-Go can all connect to eachother
-The Friends and Dream Town can connect to eachother
I'm not sure about how the remaining Japanese models connect to eachother so I think someone with a better understanding of the Japanese models will have to help you there.
-Has more to do than the ones I currently have
All the ones above have more to do than the originals! I'll list a few features from each:
Vintage releases:
Osutchi and Mesutchi: Quite a simple model as far as I know, though there's a complex breeding mechanic where connecting your Tamagotchis gets you different characters.
International releases:
V1: This one has a two games instead of the one like the originals, and the ability to connect with other Tamagotchis. This also allows Tamagotchis to gift eachother items which they can use. Overall, the number of features is still pretty lacking, though.
V2: There's new games, and games now give the player points that they can use in the shop to buy food and items. Relatively limited compared to later releases, but still allows for quite a lot of gameplay time.
V3: Lots of new features and items, and a whole bunch of games. A big part of this Tamagotchi was the Tamatown functionality, but since the website shut down nearly 6 years ago this functionality has been inaccessible. You're not missing a whole lot though, honestly, since the souvenirs from Tamatown didn't actually do anything except look nice.
V4: Growth is now determined by your Tamagotchi's interests and skills. These skills then go on to influence the job your Tamagotchi gets. There's also a huge variety of games, increased by the range of jobs your Tamagotchi can now engage in. Tamatown functionality is lost once again, but again it's not really a big deal.
V4.5: Basically identical to the V4 but with different characters and games. I think getting skill points is easier too now and if I recall correctly the jobs are a bit different? V4 also suffered from some pretty bad glitches which were fixed in the 4.5.
V5: Tamagotchis are now put into families - if you've ever wanted to raise more than one Tamagotchi at once, now you can! Though it's functionally identical to raising one. The range of features is pretty limited compared to the V4 though, since there's no job mechanic anymore. I think the range of items was also pretty limited if I recall right, and a lot of the items were only obtainable online, though the passwords are still available online since they don't always rely on a username like before.
V5.5: Similar to the V5 again. V5 had some pretty bad glitches if you were unlucky - mine liked to reset itself on occasion - and they were fixed in this iteration. Different characters and games once again.
Music Star: Basically the V4 but you can only get one job, a career in music. This isn't too much of a downside though, as there's plenty of fun to be had - more varied gameplay than previous versions, for sure. Easily one of my favourite versions, though a couple of the characters and a few of the items aren't so easily accessible anymore since the shutdown of Tamatown.
Tama-Go: Very, very limited, unless you buy the additional figures which clip onto the top and unlock more content. The figures unlock extra games and the ability to buy items, which was removed from this version for some reason. On a more positive note the screen is now a 4-tone greyscale screen instead of the previous monochrome screen, and that allows the player to buy different room designs for their Tamagotchi.
Friends: Haven't played it, so I don't know much about it. From the sounds of it though, it's pretty limited, like the Tama-Go. Connection is NFC instead of IR, now.
Friends Dream Town: Supposedly an improvement upon the Friends, but still suffers from being relatively limited once again.
I don't have any of the Japanese-exclusive releases, but I've listed some of their features here anyway:
Keitai: Kinda like the V3 but with more things to do, I think?
Hanerutchi 1: It's based off a gameshow or something? So the characters aren't really Tamagotchi characters. I think it's like a Keitai or the Plus / V1 otherwise.
Akai: Like the Keitai but the pixels are red and there's different games and characters
Entama: Like the V4, but with more to do, I think?
Uratama: Like the Entama, but the pixels are blue now. Once again different characters and games, similar to the V4.5.
Hanerutchi 2: Another Hanerutchi, but like the Entama, I think?
Oden-Kun: Not sure what the functionality is like but it's based off the Oden-kun character.
Plus Color: First Tamagotchi with a full colour screen.
Hexagontchi: Like Plus Color but with some features related to a gameshow called Hexagon, or something.
iD: I think you get the ability to keep your character for as long as you want, or something. I don't know much about this one.
iDL: Like the above but... more?
P's: Dunno, there's these "pierce" things that you can clip onto the top to unlock more content though.
4U: First Japanese model that used NFC I think. Allows the player to download new content.
4U+: Like the above but more, again, I guess.
M!x: Has the aforementioned "Mix" mechanic which allows you to mix two characters together to get something entirely new.
Meets: No idea, but I've heard the current release of this version is pretty glitchy. If a Meets sounds like your thing, maybe wait until the next version of the Meets comes out. Apparently carries the same mixing mechanic from the previous version. I think you can also raise twins in this version though I'm not sure if that was in a previous version (aside the V5) too.
Generally the Japanese models have more features than the international releases.
-$40 bucks or less (I found some on Ebay that go for over $60! Im not made of money!)
This is probably the strongest restriction you've given, and it should helpfully make it easier to make a decision. Any of the Japanese models from the Plus Color onwards will probably end up being too expensive for you. Even the Entama can go for $60+ if you're buying new, though it should be within your limits if you're buying used. Seems like the iD / iDL are an exception though, they look like they tend to go for a little cheaper. The vintage Osutchi and Mesutchi should be just within your limits. English models should all be affordable, though be careful not to buy one of the $2 fakes. The Music Star is likely to be a little expensive though some sellers will definitely sell it to you for less than $40. The Tama-Go should be just within your limits though the figures aren't going to be that cheap, and it's not really worth purchasing it without the figures. The Dream Town version might be the hardest to get because it's usually (confusingly) named the "Tamagotchi Friends" and those that know to add the "Dream Town" on the end tend to sell it for quite a bit more. On the other hand the Tamagotchi Friends seems to be one of the cheapest models available, probably because it was a relatively recent release.
Hopefully others can fill you in on the information I'm less familiar with. I hope this helps!