The "I'm pregnant" line marked the first misstep in the show's dialogue, which is saying something given that it's near the end of the sixth episode.  I will let it slide only if it makes Jo more interesting in the future.

Leon is trying desperately to make a living in a dying field as a young jockey.  There's nothing silly at all about losing a much-needed job opportunity and being crushed by it.

Hammerbutt's right insofar as that's the appropriate way to feed a horse carrots, especially since carrots and fingers can be the same size.  However, I don't think it's possible to read too much into anything Milch writes.

Re: the Four Horsemen holding out carrots for their new horse: the first thing I thought of was people lining up to take Communion.  It doesn't make a lot of literal sense, but there you go.