Is SHA-512 more secure than SHA-256?
Not in a way that matters in practice: neither has a known collision, and SHA-256's search space, 2 raised to 256, is already far beyond what any brute-force attack can reach today. SHA-512's advantage is 64-bit performance, not extra security.
Why is the SHA-512 digest 128 characters long?
Because the raw output is 512 bits, and each hexadecimal character represents 4 bits: 512 divided by 4 is exactly 128 characters. That is twice SHA-256, which has a 256-bit output and 64 characters.
Can SHA-512 be used to store a password?
Not as a final solution. Even with a 64-bit internal state, it is still a fast hash with no built-in salt; the standard for passwords is bcrypt, scrypt, or Argon2, deliberately built to be slow.