Sometimes I let offerings of water just evaporate on the shrine. Dry foods, or foods that won't spoil, might stay for a few days. Anything else I remove within a day, or even a few hours. I try to offer things that will only be okay for animals to eat, since offerings generally end up outside. Sometimes I will burn or bury them, but it's not practical for me to do all the time. Depending on what's in the cakes, they're not too bad to leave out. I have made cakes from wheat flour, honey, eggs, olive oil, and almonds, and don't feel bad about leaving them outside.
When moving shrines I don't do anything very elaborate. I like to remove the image/symbol of the deity first, and then move everything else to the new shrine, and then add the image back in. Over time you'll probably develop your own little practices when it comes to moving them. There's not really a right or wrong way, as long as it's done respectfully. Cleaning shrines isn't bad either. In Greek tradition they're generally cleaned once a month as part of the monthly festival calendar. You don't want dust to layer on the statue or other items if you never move or clean them, after all.
I sing of Artemis, the pure maiden, who cheers on the hounds, the hunter of stags, who delights in archery over the shadowy hills and windy peaks, who draws her golden bow, rejoicing in the chase.