What is the warmest place on the Martian surface? What temperatures are seen there? (day, night, seasons, etc.)
I am asking about weather; not anything out of the ordinary such as a volcano.
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This is a bit tricky to answer, as we don't have perfect global weather from Mars. But as can be seen from the images at this site, it will be a low lying area near the the equator. If I had to venture, I would say the sourthern part of Syrtis Major, but I haven't seen a high enough detailed graph to definitively answer this question.
What is the warmest place on Mars during the night ?
Probably Badwater crater (32.8S,62.1E) in the Hellas basin.
A part of its floor has the lowest elevation on Mars of about - 8200 m ! (measured with Mars Trek)
So it has the highest concentration of CO2 and thus the warmth of the Sun during a southern summer day will be kept there the best at night ?
Plate 12 of this article shows that near the northern rim of the Hellas basin there are the highest global nighttime temperatures.
Hellas, which exhibited high thermal inertias in previous TES maps, is now lower as a result of eliminating high dust opacity data,...
Figure 1 of that article shows that the Hellas basin is not the only region with high nighttime temperatures.
Also,it has been suggested that Badwater crater could be the warmest place because of the greenhouse effect, but the atmosphere of Mars is just too thin to have such an effect.
In Global distribution of bedrock exposures on Mars using THEMIS high-resolution thermal inertia 960 instances of very high thermal inertia regions on Mars were obtained.
Among the regions that are situated along -24$^\circ$ latitude should be the ones with the highest nighttime temperatures because there, in southern summer, Mars is closest to the Sun.
Figure 1 of Rock abundance on Mars from the Thermal Emission Spectrometer shows that the max. nighttime temperature in those regions with very high thermal inertia can be about -60$^\circ$C.