Nobody has a “great chance” at Harvard… but you do have a very real chance. Being an immigrant Dreamer is a hook, on top of your stellar accomplishments.
Generally summer programs don’t make much of a difference, but the MIT summer program is extremely competitive – about as competitive as getting into MIT.
If you were to apply to lots of T20 schools, you would likely be admitted to several. (Though should always have some safer backups).
As to finances… Many of the very top schools have the broadest definition of financial need, giving the most money.
For Harvard, at a family income of up to $150,000, they only expect family to contribute up to 10% of their income:
For Yale:
Families earning between $75,000 and $200,000 (with typical assets) contribute a percentage of their yearly income towards their child’s Yale education, on a sliding scale that begins at 1% and moves toward 20%.