Yes, you should still make attempts to make repairs. Even if your tenant is not paying the rent, they can make a claim that you violated the warranty of habitability if they complain to you about a repair that is needed and you fail to fix it. Your obligation to make repairs is not tied to the tenants obligation to pay rent. In other words, a tenant not paying their rent does not cancel your landlord obligations.
This will depend on quite a few factors - whether or not you have already sent a valid notice of termination, which county the case is brought in, whether the sheriff has difficulty serving your tenant(s), whether your tenant(s) make any allegations against you (or ask for a jury trial). If you have already sent a termination notice and are ready to file with the court, it is best to plan on a minimum of three (3) months. Sometimes evictions resolve faster, but sometimes they can take longer.
Every landlord-tenant relationship has a lease agreement. The agreement may be written or oral. A written agreement may have a long term or it may be on a month-to-month basis. Most written leases do not vanish after the expiration of the initial lease term, the term just shortens to month-to-month. Caveat: The status of your expired term written lease may depend on the language that is in your written lease. Please consult with an attorney to determine the status of your expired term written lease.