Definitely get a lawyer to write up all the details of the partnership in a formal agreement.
If your ex does not want to do this, that is a bad sign. You both need to be clear about expectations and responsibilities in this partnership, and define an exit strategy in the case one of you wants out. This is the most fair to both parties.
Generally, what is common is that property is split cleanly when the relationship ends.
I would strongly recommend you both work towards a clean split with no joint property ownership. How this looks depends on your unique situation.
To address your questions 2 and 3:
You have two roles here - tenant and owner.
As a 50% owner, you are running a business with a partner.
That business will have assets (home), income, expenses, and profit. You basically need to run this partnership as a simple business.
All the rent income (your rent and the other tenant's) should go into a separate account. The mortgage and all other housing expenses are then paid from only this account. Any excess is then profit that may be split 50/50. All expenses should be agreed upon by both of you, either by contract or by direct communication. You should see a financial professional to make sure accounting and taxes are set up properly.
Under this system, your ex could do work on the house and be paid from the business income. However, they are responsible to you to provide an estimate and scope of work, just like any other contractor. If you as a joint owner agree to his price, he then could be paid out of the business income. This reduces the business cash flow for the year accordingly.
You can probably see how this can get very complicated very fast. There is really no right or wrong answer on what both of you decide is fair and best. For the sake of simplicity and the least chance of a disaster, the usual and recommended action is to cleanly split all property.