Yes, but the game doesn't tell you which. As long as your Druid isn't a newborn baby, they've likely seen some beasts in their life (especially during and after their Druid training!), and therefore can Wild Shape into those beasts.
Your background is probably the best guide to what creatures you've likely seen, as your background will imply a region that's familiar to you. Any beast common to the region in which you grew up and trained is a beast you've likely seen many times, and uncommon ones are likely to have been seen once or twice too.
This is all between you and your DM, of course, because the game doesn't rigidly define a list or rigidly define a way to create one. You're two reasonable people who enjoy playing games together though, so I'm sure you'll figure out an enjoyable, reasonable set of beasts that you can agree makes sense for a nature-loving outdoorsy person to have ever seen in their life.
Your analysis is correct but one sided
You have accurately pointed out many of the advantages of the Moon Druid. It is true that it is one of, if not the strongest subclass in the game. But you have failed to account for some of the drawbacks that limit their output.
- Low AC: Most beast forms have comparatively low AC compared to the martial characters in the party. The brown bear you have based most of your analysis on has an AC of only 11. Most attacks are going to hit so you need those HP to make up for it. Additionally you will be making lots of concentration checks for any concentration spell you cast before Wild Shape. Barkskin can help somewhat but AC16 isn't that high and it means you can't concentrate on spells like Moonbeam or Flaming Sphere instead.
- No casting in Wild Shape: Yes the Druid is a full caster, but they have no access to this while in Wild Shape. You have a lot of options and heaps of resources but are limited by the action economy to only doing one thing at a time.
- Weaker Spell List: Compared to the Wizard/Sorcerer spell list the Druid has a lower damage output for pure spell casting at most levels.
Then compared to the Cleric they have access to fewer buffs and high level healing spells. I'm not suggesting this makes them weak, but they certainly aren't the most powerful full caster going around.
- No decent cantrips: When the Druid is casting, they have to use spell slots to get a decent damage output. Their cantrips are very weak compared to other full casters and they have no options to add their casting mod to it unlike clerics, warlocks, or wizards.
- Can't use combat feats: The Moon Druid is a strong melee character while in Wild Shape, but not as strong as Fighter or Barbarian. They have a low AC and can't access feats such as Great Weapon Master or fighting styles to increase their output.
- Complexity: Playing a Druid can be overwhelming. With the wealth of options and abilities available it can be hard to make effective choices. It is not a class I recommend to new players. Unless they are really keen on the Wild Shape part.
I am currently playing a Drow Druid. In my party (level 6) we have a Champion Fighter, Rogue Scout, War Cleric, Evocation Wizard. I have been tracking the kills and damage output in combat since the start of our campaign. Early on, I was slightly in front as Wild Shape gives a huge bump at 2nd level. As the campaign continued however the Rogue quickly overtook me as she increased her damage output with additional sneak attack dice. After gaining Extra Attack at 5th level the fighter has also surpassed me in damage output. At 6th level and gaining a magic weapon the fighter in by far our highest damage dealer.
My typical combat strategy is to combo Moonbeam and Brown Bear Wild Shape (Polar Bear at 6th level). This is pretty good until I lose concentration, which usual happens in 1-2 rounds due to my low AC (lots of checks) and no proficiency in Con Saves.
One session our Cleric was missing and I was forced to drop Wild Shape to heal my allies. This is evidence of the "can't do it all" limitation of the 5e action economy. While I end most encounters with more HP or spell slots than most of my party I'm not able to use all of my abilities together.
This opens up new Wild Shape forms for the Moon Druid for the first time. CR2 forms are a big step up from CR1 and will address the reduced power at level 5. The best damage combat form at this level is the Giant Elk. 4d8+4 damage against prone targets is nothing to be sneezed at. There are other great options such as Giant Constrictor Snake but they all suffer the same low AC.
However you only get one attack per turn and still have a low AC of 14. Meanwhile the wizard now has multiple castings of fireball, the fighter gets a second ability score increase and Cleric get a second use of channel divinity.
After playing at 6th level for a while I can say that my damage output still lags behind the fighter and rogue. This is mostly due to a lower to-hit modifier and a lack of extra attack. The additional hit-points of forms at this level make me a more effective tank, particularly now that I have enough spell slots to burn for healing. Combats usually involve me targeting a single enemy and tanking them out of the fight while the others deal with the rest.
Are they OP?
The Druid is likely the most versatile class in the game. They can be a competent healer, decent melee character or a pure damage spell caster. However they can't do it all at once and will never be quite as strong as a character that focuses on one area.
The Moon Druid Wild Shape is possibly OP at level 2. It is a huge bump in damage and HP compared to what others receive at the same level. It does tail off within a few levels though as other classes gain combat features. The bump at 6th level keeps you on par with other classes but is the last major improvement for a while.
Overall I'd say it is strong, maybe the strongest but not so much that it is OP. Any correction to weaken the class risks making it unplayable by taking away the flexibility that makes it so much fun to play.
The Cranium Rat has an "always available" bonus action, but I would not recommend it for combat. It is a tiny CR 0 beast from Volo's Guide to Monsters and all it can do with its bonus action is this:
I believe it is the only beast with an "always available" bonus action at the time of writing.
Homebrew and UA aside, the only ways to get an "always available" bonus action for Wildshape are to multiclass into a class that grants you a bonus action (such as the Rogue), or to be one of the few playable races with a bonus action (such as the Goblin)... though of course it is too late for your character to be a different race.