Does it make sense to mention months (and days) in any BCE date?

No, you should not exclude months and days from your timeline software.

While the posting of the year is absolutely necessary when accurately chronicling an event in historical time-(especially if such an event occurred, "Before the Common Era'), the month and even the day, should also exist alongside it.

I don't necessarily think that months and days must always appear alongside the BCE year, though if it was a landmark event which transformed human history, we, as students of history, should be aware of its chronological positioning. 

Display it to the available accuracy of the event.

Does the event have an accurate month and day? Yes. Whether you choose to display it in modern calendar dates or a contemporary historical one is another matter.

Does it not? No.

For example, the reign of Pharaoh Menes is disputed, but perhaps between 3200 and 3000 BCE. What's what should be displayed, along with the note about the questionable accuracy of the start, end, and length of his reign. (You should definitely mention that he was killed by a hippo.)

Quirks of date parsing and internal representation are an implementation issue. Unix time may not be the best solution; it represents a specific moment in time. While this may be convenient for calculations, it is not good history. You may need a more complicated data structure which can store the inaccuracies of historical dates.

An analogous problem in engineering is significant figures. For example, if I measure with ruler only accurate to 1 cm that a board is 40 cm long, then I cut it into thirds, I get 13.3333... cm. This gives an impression of precision which simply does not exist; you're only as accurate as your least accurate measurement. You measured 40.??? cm. If you cut that into thirds you have three boards of 13.??? cm.

It's similar with historical dates. Let's say you have "in the year 2023 Emperor Soinso did blow his nose with a mighty noise". If you store that as 1672531200 (2023.01.01 00:00:00) you're implying an accuracy which does not exist. Your date is 2023.??.?? ??:??:??

Finally, as noted in the comments, be careful with negative years. There is no year 0 (unless you're an astronomer). When systems do allow negative years, year 0 is usually 1 BC, year -1 is 2 BC, and so on. 3150 BC is -3149.

Date.parse is only consistent using the ISO 8601 format, otherwise it's browser dependent. ISO 8601 format allows for the representation of reduced precision making it a better choice for internal representation, and it is easy to compare and sort.

This is a complicated question, but the bottom line is "it all depends."

The Romans had a month of March which had an Ides which was the 15th day of March. What is the relationship between their March and ours? Between the Ides of their March and the 15th of ours?

The names line up; Our March is clearly related in some complicated way to theirs; they both happen in roughly the same time of year...but... Our March under the Gregorian calendar happens at the same time each year (modulo the leap-year dance that shoves it back and forth over the span of a day). The time span between any two March 15ths will be an integral number of years to within a day.

But the Gregorian March does ''not'' happen an integral number of years after a late-Julian March -- the Julian March had drifted with respect to the seasons by up to ten days. And when you get back to Republican Rome's calendar the relationship between our March and theirs is vague and can change drastically from year to year. (Partly because their year wasn't of any particular length, and sometimes they stuck in extra-monthly times or extended or trimmed the length of some months. When they got around to it...)

Our March is the third month of the year. It wasn't always. Theirs was too...sometimes...

A good method for handling this is to convert dates into Julian Days (unrelated to the Julian calendar!) which are used by astronomers. Julian days which is a simple count of days since January 1, 4713 BC. (This was selected to be far enough back that there would be no records with negative Julian Day dates.) Wikipedia has a pretty good discussion of this.

There are libraries in most major SW languages to convert between Julian Days and most major calendars. You can use this to get accurate intervals, to get an absolute ordering of events (provided you know their dates accurately enough) and to compute durations.

But it all depends on what you're doing. If you are, say, narrating Caesar's life and death, you can use the chaotic Republican Roman calendar's days and months and segue over with barely a footnote to Caesar's reformed Julian Calendar. In that context, speaking of "March" makes perfect sense.

But if you're talking about Roman climatic patterns, this simply won't do. Nor will it do if you want to synch Caesar's Ides of March up with events elsewhere or elsewhen. You must dig into the gory details and give up on treating March SPQR and March USA as the same thing.

Not a historian, but yes it makes sense. Months are associated with certain seasons and astronomic phenomena (equinoxes, solstices) and months with days can help with orientation if several events happen in quick succession or in parallel. Even without days, months are useful as a general reference.

On the other hand giving an exact date can give a wrong impression of precise knowledge. E.g. people sometimes give an exact birth date for Genghis Khan even though Mongolians were notoriously unaware of their birth dates until quite recently and although the relevant sources can not even agree on a year.

One very famous date BCE are the ides (the 15th day) of March 44 BCE. So dates before CE do not only make sense, people actually use them. I just had a look at several books on Greek and Persian history, and they also frequently use months and year (e.g. October 333 BCE), although some clearly prefer to just mention the season and the year.

A common alternative is using a local contemporary calender. E.g. in texts about East Asian history it is common to use dates like "the third month of the 8th year of the Zhentong emperor". This is closer to the original sources and avoids calender conversions which may be problematic in some cases. But it also is much harder for people who do not know what the reference date of that specific calender is.