Pokemon Go – Currently Obtainable Pokemon from Previous Generations

Your situation sounds a little bit unusual, but you can increase your chances of catching a Pokémon by waiting for the inner circle to be as small as possible when your Pokéball hits the Pokémon. According to the official Pokémon Go catching guide:

You have the greatest chance of capturing the Pokémon while the colored ring is at its smallest diameter. At the opportune moment, fling the Poké Ball toward the Pokémon.

There is travel time, so don't wait too long or the ring will become maximum size again. Also note that this allows you to get "great" and "excellent" throw XP bonuses, though you have to be particularly good at aiming to make your throw land inside of the ring when it is so small.

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.