From my own personal experience you cannot receives eggs from pokestops if your item inventory is full even if you have less than 9 eggs in your pokemon inventory. Some cursory searching doesn't show anyone who has been able to receive eggs when inventory is full.
Source: Personal experience and here
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.
Best Answer
You can obtain eggs from pokéstops only when you have free bag space (when you actually receive items from the pokéstop) even when you have empty slots for eggs. I usually discard revives to stay slightly below the cap until I have 9/9 eggs again.
Source: personal experience