We are getting a hot plate for Shabbos. On Shabbos, how does one reheat cold food from the fridge on a hot plate? Is it permissible for a Goy to put cold food from the fridge onto the hotplate to reheat it? Our hot plate does not have a timer yet or a temperature adjuster.
Hotplates are permissible to use for Shabbos if used in a kosher manner, i.e. being careful about placing cold food on the hotplate (Nesina Batchillah), and how to return pots back unto the hotplate (Chazara).
Placing food before Shabbos (Shehiya): If the hotplate has no adjustable controls, this is permissible according to most Poskim. Still, one must be careful not to tamper with the food on Shabbos in a manner that hastens the cooking (Maigis, Kiruv bishul). [There is a minority view that forbids placing food before Shabbos, even on non-adjustable hotplates, unless sufficiently cooked or if an extra Blech or silver foil etc. is used on the hotplate.]
The Badatz of Crown Heights has authorized a hotplate for Shabbos, produced by Tech Yid, which meets even the strict requirements. Most important is the fact that it has been produced with safety as the highest priority and has been thoroughly tested for this.
See here and here.
Bishul and Nesinah Batchillah (placing food to be heated on Shabbos): Placing any food which was not completely cooked, on the hotplate on Shabbos is a violation of the Melacha of Bishul. Furthermore, even completely cooked food that has cooled down may not be placed on the hotplate either to reheat, due to the Melacha of Bishul, unless the food is completely dry (and will remain dry when heated).
Some Poskim allow placing hot food or fully cooked and absolutely dry food directly on a hotplate on Shabbos, though some are Machmir (stringent) to place the food only on top of a pan already cooking on the hotplate. The same applies when done via a non-Jew.
ראה הנסמן במ״מ וציונים להלכה יומית אות תעח בתחילתו. שם אות תשנג.