A comma is usually required before 'and' when joining two independent clauses, but it's not always mandatory. Typically it can be omitted when the independent clauses are short and closely linked in meaning. 'How short is short?' is a matter of judgment; I would say that the clauses in your example sentence might be a little long, but they are close in meaning, so it's something of a toss-up.
Removing the second 'he' would turn the second clause into a dependent clause, and thus a comma would be unnecessary. "He walked down the street and then turned the corner."
A comma is definitely necessary if you eliminate the 'and' but still want to add a dependent clause. "He walked down the street, then turned the corner."
Here is a very clear explanation page: https://www.grammarly.com/blog/comma-before-and/