Is the generator G of the keypair always the same number for all different tx messages in Bitcoin? If so, what is that number and why can't you divide the public key by G to obtain the private key?
No sufficiently efficient algorithm for division is known. Multiplication over a closed group is a lossy process.

For an imperfect analogy, consider multiplication over the group of 100-digit numbers where you just keep the last 100 digits of the result. Since you don't know what digits were discarded, you can't naively reverse a multiplication.

The corresponding algorithm in DSA is modular exponentiation -- (G^X) mod n, which is also, for practical purposes with large numbers, irreversible even if G and n are known.