Can somebody take the transaction tx0 I signed and publish it in order to spend the new coins with the old transaction?
Firstly, the transaction inputs are already known to have been used. Any node examining the block will know that the rerun transaction has invalid inputs.
Secondly I expect (but haven't checked) the inputs to the signature include both addresses. This prevents the signature being misused in the way you suggest. (See Murch's clarification in comment below)
It may help (if not you, some other readers) to remember that bitcoins don't exist outside or inside the blockchain. The only thing that exists are transaction inputs and outputs. You can't use a signed transaction with inputs A,B and C to use inputs D and E. The signature prevents you altering the signed contents without invalidating the signature.
how is it that address reuse is not directly forbidden? if not, can you detail why?
Addresses can be reused. The most obvious problem is one of privacy. If you give someone an address so they can buy $0.01 of tent-pegs from you, they can then see on the blockchain that you also received $300,000,000 recently to that address and go buy a big hammer and come to visit you at 3 a.m.