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Since you feel there's no reason for your world to have the same days of the week as our world (that's reasonable), why must your world have weeks at all? Why must the weeks be of X days? A month is a length of time that's tied to a natural phenomenon - the turn of the moon around the earth. The week is tied to nothing but religion. Is there a similar justification for it in your world?

You can, if your story demands it, make some justification for having weeks, you can pick the length of those weeks, you can have weekly day(s) off, you can choose the names of the days (although some cultures just have names that mean 'first', 'second' etc., so that too is an option).

But consider first - why do you need this at all? What does your story gain, that would not be expressed by "on the tenth day of the seventh month"? If your story would gain something by adding day names, go ahead and add those. If it gains nothing, don't bother. ("The tenth day of the seventh month" is how the Torah sets up Yom Kippur, btw. So that's a real-life example.)

I don't see why. Presumably your book is written in English, but people in your world did not speak or write in English, so you offer a translation of the events. Why would you translate everything else except the words for the days and months?

It would only make sense if there are concept which are not readily mapped to words known to the reader, for example if there are extra days in a month not belonging to a week, or if there are a different number of months/days completely - maybe there are two moons or two suns.

You are creating a world and a society. Have faith in your choices and create your own names for months if you wish. What reason would your world have for having months named after Roman emperors?

You could have your world follow a lunar year as that is more intuitive. New month, new moon.

It is your world. As long as the paradigm holds, go for it. You can even rename the seasons, the reader will know that a harvest season is probably late summer and fall. We have artificial divisions of time based on religion, the industrial revolution and emperors your world never knew.

One could say that authors need to be pretentious to carry off the world crafting and give the characters an appropriate setting and culture.

Make it different and have fun doing it.

You don't need to invent new names for concepts, unless they somehow relate to the storyline.

Instead, use relative names like tomorrow and yesterday, but not Thursday or next-week. Relative Celestial names like "next month" or "after the new moon" or "next growing season" are totally transferable too.

+1 for realising its pretentious. Remember the Fiction Rule of Thumb in https://xkcd.com/483/

I would say No for the most part. There is the question of "Why would we have a Saturday if no one's ever heard of the god, Saturn" or indeed Rasdahan's point of naming months after Roman emperors.

However, there's a known trope in writing, the "Translation Convention", which means that even though the characters in their universe aren't speaking English, it's presented to the reader in English and the reader is just not supposed to think about it too much.

Most words in English - and presumably all languages - has an etymology that is specific to our world, so days of the week or months of the year need not be any different. You'd end up translating most of the dialogue if you took all the Latin-influenced words out (for example) because Latin didn't exist in your world.

However.

Perhaps your world also has gods (or highly honoured faeries, or heroes or species of insect or something else) as part of the story or history of the world, and as part of enriching your world you want the people to have named time periods after them. As a deliberate thrust in that direction, then it would do no harm and put some depth into the world (but keep it to a minimum, you don't want the reader to have to be memorising the entire new calendar). If you're doing it for that reason, fine, but don't feel like you have to simply because "there are no Norse gods in this world, so they couldn't have a Thursday"

I'd say no. George R. R. Martin managed to write five really long books without having to name a single day. He used "third day of the seventh moon" and stuff like that.

So let me pose this questions. Using only the Star Wars films, describe for me the physical shape of a Nerf that would be herded by "Scruffy-looking Nerf Herders.

The point here is that you may have a date system that is different to our world for various reasons (Your world has a 28 hour day cycle... or a 422 day annual cycle... or is like real world time, but the Norse Gods don't exist so they can't lend their names to the 7 days of the week). This shows a really decent amount of world building and attention to details that fans will appreciate... but how often do you name the day of a week in the fantasy genre (Harry Dresden and Potter might, because both are in genres that days of the week could matter on (the former is Fantasy Noir, and thus might comment on how it's an unusual Wednesday and the later is in a Fantasy Boarding School genre and will be in certain classes on certain days or off on the Weekends))?

A casual drop of an unusual day name in a common Euphamism that will translate to the readers (i.e. A talking camel declares Scigduleday his favorite day of the week, because it's Hump Da). This will immediately translate to the audience because the joke is referring to the famous Geico Commercial that features a talking Camel who is just so happy that it's Wednesday. The reader can infer that the date system is different in some way, because of the way the day is used in conjunction to our real world days. Scigduleday is clearly the middle of the work week. Thanking the Gods that it's Farramday will instantly line up with the TGIF of real life, and that Daantday Morning blues is clearly the beginning of the week.

It is not necessary for the reader to understand the dating system to get the joke that the days might be different but the sentiments about them remain. You can overbuild the world and show that there is constitantcy, either by making the jokes to show you thought about this, or if you have a spinoff, properly dating things (The Stardate system of Star Trek came in handy when it's consistent use in the TNG era was helpful for fans to know Chronology of the events that occured over two seperate television series and a movie franchise at it's peak in the 90s (TNG films, DS9, and Voyager all were in production at this time. The last episode of Voyage happens shortly before the last TNG movie (within a year of each other)).

Showing you created the system is what can be termed as Biblical information, which are documents and notes that one can use as rules for how the series operates. For example, J.K. Rowling first "outed" Dumbledoore during production of the Sixth movie as the original script had Dumbledoore refer to a young romance he had with a woman... because she had this information thought up for some time, she got the line removed from the script. This was sometime before she gave the more public outing. Similarly, the team of How I Met Your Mother, had to do several Biblical notes as the series progressed and they realized they needed to keep things consistent. They had had enough forwarning to shoot a scene in season 2 that would only be used in the season finale, which at the point in there production run, was not known to them (it was indeed used in the season 8 finale for the series).