Added In answer to the question: where are the sources about piety? Unfortunately I can't quote specific sources. One was from Talmud Sukkot as analyzed by Professor Moshe Halbertal in a class I took. He discussed this specific issue and referenced other Talmudic sources on the issue too. Another example is the mitzvah of the meal before Yom Kippur--thus limiting the length of the fast. There are others too. Remember that when the sources explicitly limit something, they're often saying that it is not better to do "more."

While researching the questions I asked, I found that there is a wide variety of practice on the question of re-heating solid food on Shabbat. So the answers really depend on you and your community.

Here are some answers that I've found from different Rabbinic sources on the Internets:

Warning Don't take halachic advice from me! Consult your Rabbi

Sources:

1. Rabbi Asher Lopatin: The Dos and Don'ts of Re-heating on Shabbat. R. Lopatin is an Askenaz modern Orthodox rabbi in Chicago. I found this source to be concise and helpful.
2. Rabbi Harvey Well: Cooking and Reheating Food on Shabbat
3. Rabbi Shlomo Levi: Analysis of Rabbi Ovadia Yosef's lenient stand on re-heating cold food by using a plata R. Ovadia Yosef is a Sephardi Gadol (great rabbi) in Israel.

My personal conclusions from the above sources are:

• The food we're talking about here is solid and was completely cooked before Shabbat. The plata was activated before Shabbat (turned on or its timer was set).
• Can I use it to reheat solid food from the refrigerator on Shabbat? Yes. Rabbi Asher Lopatin says so clearly, source 1. Rav Ovadia Yosef is reported to say it's ok (source 3).
• Do I need to put an empty pot/cooking tray between the plata and the food container? This is an extra stringency which many follow. And many do not. So check your rabbi and community for your personal answers. R. Lopatin does not require. If I read source 3 correctly, R. Ovadia Yosef doesn't either.
• I can move the food to the plata on Shabbat morning, right? Yes.
• Any other issues with using the plata? I can't determine this from the sources.
• Should I not stir the food while on the plata or does that issue not apply? I can't determine this from the sources I found.
• I can't reheat soup or other liquids on the plata, right? Definitely cannot. Only "solid" foods, foods which are not "poured." (Some amount of gravy can be on the chicken.)

I found R. Lopatin's instructions of the most help. As I note above, he's firmly in the Modern Orthodox camp. It is interesting that Rabbi Ovadia Yosef is reported to have given the same opinion re plata use.

Bonus answer: Is a plata the same as a blech? Definitely not. A plata is like a Kdeira Blech (a water blech). But a plata is not the same as a Kdeira Blech. R. Lopatin treats the two the same (source 1). But some do not--that's where the idea of an extra pot on top of a Kdeira Blech or, perhaps, plata comes in.

Bottom line This issue of reheating cold food on Shabbat (that was cooked before), is a specific area of leniency/stringency that different rabbis and different communities have their own opinions on. No answer is "right." Rather, each community's answer is "right" for that community.

Remember that Judaism does NOT say "the more piety the better." Don't make the mistake of thinking that greater stringency is better. What matters is your community's answer to these questions.

While researching the questions I asked, I found that there is a wide variety of practice on the question of re-heating solid food on Shabbat. So the answers really depend on you and your community.

Here are some answers that I've found from different Rabbinic sources on the Internets:

Warning Don't take halachic advice from me! Consult your Rabbi

Sources:

1. Rabbi Asher Lopatin: The Dos and Don'ts of Re-heating on Shabbat. R. Lopatin is an Askenaz modern Orthodox rabbi in Chicago. I found this source to be concise and helpful.
2. Rabbi Harvey Well: Cooking and Reheating Food on Shabbat
3. Rabbi Shlomo Levi: Analysis of Rabbi Ovadia Yosef's lenient stand on re-heating cold food by using a plata R. Ovadia Yosef is a Sephardi Gadol (great rabbi) in Israel.

My personal conclusions from the above sources are:

• The food we're talking about here is solid and was completely cooked before Shabbat. The plata was activated before Shabbat (turned on or its timer was set).
• Can I use it to reheat solid food from the refrigerator on Shabbat? Yes. Rabbi Asher Lopatin says so clearly, source 1. Rav Ovadia Yosef is reported to say it's ok (source 3).
• Do I need to put an empty pot/cooking tray between the plata and the food container? This is an extra stringency which many follow. And many do not. So check your rabbi and community for your personal answers. R. Lopatin does not require. If I read source 3 correctly, R. Ovadia Yosef doesn't either.
• I can move the food to the plata on Shabbat morning, right? Yes.
• Any other issues with using the plata? I can't determine this from the sources.
• Should I not stir the food while on the plata or does that issue not apply? I can't determine this from the sources I found.
• I can't reheat soup or other liquids on the plata, right? Definitely cannot. Only "solid" foods, foods which are not "poured." (Some amount of gravy can be on the chicken.)

I found R. Lopatin's instructions of the most help. As I note above, he's firmly in the Modern Orthodox camp. It is interesting that Rabbi Ovadia Yosef is reported to have given the same opinion re plata use.

Bonus answer: Is a plata the same as a blech? Definitely not. A plata is like a Kdeira Blech (a water blech). But a plata is not the same as a Kdeira Blech. R. Lopatin treats the two the same (source 1). But some do not--that's where the idea of an extra pot on top of a Kdeira Blech or, perhaps, plata comes in.

Bottom line This issue of reheating cold food on Shabbat (that was cooked before), is a specific area of leniency/stringency that different rabbis and different communities have their own opinions on. No answer is "right." Rather, each community's answer is "right" for that community.

Remember that there are many sources in Judaism which argue against the concept of "the more piety the better." These sources would urge you to not make the mistake of thinking that greater stringency is better. What matters is your community's answer to these questions.

On the other hand, there are some who do feel that "the greater the piety the better."