Not really, no. The only Pokemon that would be considered to be 'only hatched from an egg' are the region-restricted Pokemon. But, even then, if you were to travel to one of those countries, you could of course catch those as well. The region-restricted Pokemon can be seen here: What are the regional Pokemon?
Edit: I feel it is important to add that this is still a pretty hotly debated fact. There was a survey done which claims to conclude that out of approximately 40,000 hatched eggs, none of the Pokemon were region-locked Pokemon. However, there are some people who claim (with evidence) that they have hatched region locked Pokemon. Some of these people even claim to have spoken to the person who created the aforementioned survey, only to have that person dismiss their claims as lies or 'hacking'.
So, right now, I am leaning towards that it IS possible to hatch region-locked Pokemon, however, there is yet to be 100% conclusive evidence either way. You can see some of the threads on Reddit where users have made claims for both sides.
Edit 2: Just to be clear, the answer to the actual question at hand here still appears to be no, the edit above is just some additional information, some of which may be a bit beyond the scope of this question.
If you can see the Gym, you can see if there's a Raid Battle going on. If you can't see the Gym, you also won't see the Raid Battle.
"A Raid Battle is About to start nearby" refers to the Gyms which you can see. (just like above)
Best Answer
You can do the raid with the full Pokémon box. You cannot however capture the raid boss afterwards directly; you'll get a message saying your box is full (and if you haven't upgraded to 1000 yet, to upgrade in the shop). Then it will kick you out to the map (unless you click "Go to upgrades" in which case it kicks you to the shop, but either way the raid capture is done for now).
If this happens, you may still be able to capture the boss. You'll have to first transfer a Pokémon or upgrade your storage. You of course can instead ignore the raid boss if you prefer.
Then, assuming the boss hasn't left the gym yet (i.e., the timer hasn't expired), and assuming you're still "in range" of the gym, you can tap on the gym, and it should send you directly into the capture interaction. This does seem to have some bugginess in it; you may or may not be able to know the CP you're getting (there were some bugs there with re-entering showing you the wrong CP - see issue 12 here) and sometimes you get "Error" and can't do it. In my experience I've never had an issue re-entering, but some people have issues regularly.
Note that this only affects the Pokémon storage; the raid items will always be delivered, letting you go well over the maximum item storage.