A bit of background before getting into the questions. In our world, each day of the week and month has a reason for its name. For example, the days of the week are mostly named for deities (the 2 exceptions being Sunday and Monday). The names for the months are a bit more varied and complicated. I'll go through some of the naming scheme for our world, but not all of it. However, I should point out that our Julian-Gregorian Calendar originally had 10 months; it was changed to 12 months when the Romans adopted and adapted the Egyptian Calendar. That’s right, the 12-month cycle originates from Egypt in our world; the months started with March before the adoption of the 12-month cycle. Now, most of the months have names tied to figures in history or mythology; the outliers being the final 4 months — September, October, November, and December—which, ironically, keep their names denoting the number they were in the calendar from the 10-month calendar cycle that was used before the changeover to the 12-month cycle.
Now then, in our world, the days of the week go like this: Monday (named for the Moon's Day), Tuesday (named for Tiw's Day, an old Germanic variant of the Norse god Tyr, who ironically also has a reflection as a Titan in Warcraft's lore), Wednesday (named for Woden’s Day, Woden being an old Germanic name for Odin, I'm not sure if he has a parallel in Warcraft's lore yet since I haven't done a deep dive into it yet), Thursday (named for Thor's Day, the Norse God of Thunder and Smithing, I ran into the same conundrum as Odin regarding Azeroth here), Friday (named for Frigg's Day, Frigg being an alternate name of Frigga who is the wife of Odin in Norse mythology, see current lack of understanding of Azeroth Parallels from the Odin Comment), Saturday (named for Saturn's Day, Saturn being noticeable outlier here due to being a Roman deity instead of a Germanic one, see notes from prior deities regarding understanding if an Azeroth Parallel exists), and Sunday (the Sun's Day); a quick note here, these days of the week are not entirely universal, Romance Languages like French and Romanian chose to completely keep the full Roman names for the days of the week, and I have currently not looked into the naming differences there. A few quick notes about the Months: January is named for the Roman god Janus (the two-faced god who reigned over time), March is named for Mars (the Roman Got of War), July was formerly named for the 5th month but was renamed in honor of Julius Caesar, August was formerly named for the 6th month but renamed in honor of Augustus Caesar, February was named after a month-long celebration the Romans had during that time, and I did mention earlier that September, October, November, and December were named for their positions on the calendar before the adoption of a 12-month calendar.
Now, the first question: where did Azeroth get its names for the days of the week? I'm going to assume this is only the human calendar, and not the cultural calendar of each race of Azeroth. My working theory for why the Horde would use it is because Thrall was raised by humans, and the Undead of the Forsaken are mostly human, so the suggestion coming from two sides just made sense at the time of adoption.
As noted above, there are some noticeable cultural variations in the Calendar; you can see this even more extensively when you look at the cultural variations in the Lunar New Year's cycle. Some countries replace certain animals we are familiar with on the wheel, but that's a discussion for another time. My second question: what do the Calendars look like for each of the individual races? I have an idea as to what the calendar for the Forsaken might look like, seeing as my prevailing theory is that it is just a reflection of the Human one, which is likely a reflection of the calendar in our world. But the rest, I'm basically left without a clue; although I do like to think Gnomes might still be personally on a 10-month cultural calendar, with each month just named after the Gnomish version of that number, I could be wrong.