For the composition of alicorns, we have a comment from Lauren Faust, and the Twilight Sparkle and the Crystal Heart Spell novel.
Lauren Faust said that Celestia embodies the traits of all three kinds of pony, so word of god gives alicorns any kind of magic that Earth Ponies have. However, remember that Faust is not longer part of the show's production, so her views on the lore may or may not be maintained going forward.
Twilight Sparkle and the Crystal Heart Spell is the source of the "good, true heart of an Earth Pony" you mention in the question.
For the power of Earth Ponies, the only evidence we have is word of god from Faust:
While not magical like the other pony types, Earth Ponies do have a connection to the land that other types do not.
There has been no direct evidence of "nature magic" in the show or supplemental media. And again, comments from Faust should be taken with a grain of salt going forward, as she is not part of the production team any more.
For Earth Ponies becoming alicorns, we have no evidence one way or another. All alicorns thus far depicted have either been:
- Celestia and Luna, for whom we do not have sufficient history to comment
- Cadence, who has no real history in the show or comics, but was born as a pegasus according to Twilight Sparkle and the Crystal Heart Spell. (It is worth mentioning that in the comics, Cadence is already an alicorn by the time she meets Shining Armor in high school.)
- Twilight Sparkle, who was born a unicorn