Rating seems to affect how much your sim is paid and the size of your Sims’ tips. If they do not, then they don’t get paid and their rating goes down.
If they successfully complete the job in the allotted time frame, they will be paid and their odd jobs rating goes up. One of our Sims found two Eyes of Sulani while cleaning the beach, and those did not count towards the job. Note that this seems to mean that you have to find “trash” for it to count. For example, There is a beach cleanup job where your Sim has to successfully clean a beach in Sulani of trash three times. Once your Sim has taken one of the active odd jobs, they will be given a task or tasks and have a certain amount of time to complete it. There are several active odd jobs we have found so far, and while Sims in all neighborhoods can take the odd jobs, all the active odd jobs take place in Sulani, so your Sim will have to travel there if they do not already live there. Some jobs are rabbit holes, but some are active. The highest skill requirement we have seen so far is 5, and many are one or two points in the applicable skill, or even zero skill required. From what we have seen so far none of the jobs require high skill. The jobs may have recommended skills – the wingman job takes charisma, for example.