May one re-heat items that were baked (i.e. challah, cake, etc.) on a hot plate on Shabbat?

Shalom, It is permitted to reheat fully cooked solid foods on Shabbat because it is not considered as "cooking", which is forbidden on Shabbat, but only re-heating. (The laws of liquid foods are more strict). Therefore you may re-heat the challah and cake - however one may not heat them in a way that looks like cooking. There is an argument between Rabbinic opinions today if heating food on a hot-plate looks like cooking or not. If the hot-plate has no knobs to change the heating level (or if one covered them before Shabbat), many Rabbis permit heating the food up directly on the hot-plate. Other Rabbis forbid this, and allow one to heat the food up only on top of another dish that is on the hot-plate (and if no other dish is there, one may place an empty dish on the hot-plate for this purpose even on Shabbat). You should follow the opinion of your local Rabbi, or local community. If you are unable to ascertain their opinion, I advise you to follow the lenient view and heat these completely cooked solids directly on the hot-plate