I had a hard time formulating the question, but I wonder if a month with 31 days has a specific name, Likewise months with 30, 29 and 28 days.
I've never heard any terms that fit your criteria, so I'd say these are not common words that most people would understand. But the Wikipedia article about months states that the lunar months (as determined by the moon) are 30 days or 29 days, which are respectively called a full month and a hollow month. However, since lunar calendars are difficult to reconcile with the seasons and the sun, the modern (Gregorian) calendar is a solar calendar and its months do not really have anything to do with the lunar cycle. In that case we have 31- and 30-day months, and the allocation of days is arbitrary and historic.
No,the only thing related which has name is "leap year" which means year where February has 29 days.
I would gladly be corrected, anyhow you might be interested to read this, as it describes how the days got "distributed" between the different months:
Note: The year of twelve non-lunar months as we know it originally started with the year 45 B.C. by decree of Julius Caesar (whence the "Julian" calendar). He adapted it from the twelve-month Egyptian solar calendar, which itself actually began with the rising of the "dog star," Sirius (in the "dog days" [diês canîculârês] of July 23 to August 23). The Egyptian calendar's twelve months each had 30 days, and at the end of the year were added five days (six, in leap years) to make a total of 365 (or 366) days. Julius Caesar took the extra five or six days and distributed them among the twelve months. At that time the months of Quîntîlis and Sextîlis had only 30 days apiece. Later, in slavish political adulation of Caesar, the Roman Senate not only changed the name of the month Quîntîlis to Jûlius, but took a day away from Februârius, which had 30, and appended it to the then-30-day-long month of Jûlius, thus making his month equal to those with 31 days. Later, the Senate repeated this action in the case of Jûlius' nephew, Augustus Caesar, taking yet another day away from the "unimportant" month of Februârius and adding it to the month of Augustus, which it had newly renamed from Sextîlis. Thus it came about that today February has only 28 days normally and 29 in leap years.
Before Caesar's decree, the Roman year had had only 10 months, beginning with March, and the days between December and March had not been organized into separate months. (They had been merely a time of winter quiescence and religious festivals.) The "Gregorian" calendar was a small but important correction of the Julian calendar made in 1582 by the astronomers of Pope Gregory XIII. And, finally, the week is an approximate quarter of the lunar cycle of 27.3 days. (It has no provable historical connection with the Hebrew mythology expressed in the first chapter of the biblical book of Genesis.)