May one heat up liquids or solid foods with a lot of liquid if they are placed on top of solid foods or an empty pot on a hot plate? What if the intention is to only take the chill off of the food from being in the refrigerator and not to full heat them?

Placing cold liquids on the hot plate is forbidden because you are actually cooking. However if the liquids are luke warm (even though they are below yad soledet), it is permitted for Ashkenazim. The idea of placing on top of empty (or filled pots according to some opinions) is to get around the problem of Michze K'Mvashel. I.e Doing an act which seems as if you are cookinig. Though if in this case you are actually cooking, then this is not helpful. Even though there is an opinion that if the solid is of primary importance and the liquid is of secondary importance then the secondary becomes nullifed to the primary even when it comes to cooking on Shabbat. However, most Poskim do not rule accordingly and if there is a lot of liquid alongside the solids it is forbidden to heat up. In regards to placing cold liquids near a heat source to take off the chill - You are permitted to place the liquid in a place where you know that there is no way that it will reach the heat of 'Yad Soledet Bo', though in a place where there is potential to reach such a heat it is forbidden to place it there even for a moment. May the laws of cooking on Shabbat act as vehicle for experiencing Shabbat in its fullest and best way! Rabbi Baruch Kitay