In English, are words like 'English,' 'Monday,' and 'January' considered common nouns or proper nouns?

Since you asked for a linguistics answer, here goes.

First of all, it's incorrect to assume that linguistics assumes that there are proper noun and common noun categories to begin with, that all languages contain and distinguish between. Lay ideas of grammar don't necessarily correspond to linguistic ones.

A linguistic analysis of proper nouns proposes that they:

• describe a class to which they can be applied;
• but do not describe properties, which, if possessed, identify something as a member of a class.

This distinction is a little messy with abstract concepts like months or "week"days (I quote "week" because days aren't themselves abstract, but the concept of being an arbitrary member of a week, is.)

To quote from Alex B.'s answer on Linguistics.SE (Dublin being considered a proper noun):

"In other words, the extension of Dublin is a singleton (or there might be more elements in that set, if there is more than one Dublin). However, Dublin has no intension at all - there is no property of "Dublin-ness" that all Dublins would share."

In other words, even though "Dublin" would usually refer to Dublin, Ireland, there's also Dublin, Ontario, Canada, and Dublin, Texas, USA, and many others.

However, this is no property all these Dublins share that make them a member of the class Dublin, in the same way that all apple trees would be a member of class apple-tree, by virtue of having the property of growing apples. They are considered "Dublin" on an ad-hoc basis, and this is what makes "Dublin" a proper noun.

So we can break this down into your three questions (languages, months, and weekdays.)

Languages

This one is very tricky, and I'm frankly not sure I have an answer here. Languages can be described by properties which would identify you as a member of a class, i.e. you could look at the language someone in Frisia is speaking and decide whether it is English or Frisian.

Therefore, I propose that the capitalization of "English" is not due to its status as a "linguistic proper noun," but rather because it is an extension of English's capitalization of country/people names (from which language names are often derived).

"England" is capitalized because it is a proper noun; you can see that there is no property that makes something "an England," there is also an England, Arkansas, USA and an England, Germany.

Now, this doesn't hold for every language (there is no Hinduland that Hindi is named after), but it's easy to see that the rules for capitalizing languages deriving from the names of a country (England) or a people (the Angles/English) would be applied to languages newly added to the English lexicon.

Months

This one's easier: there is absolutely nothing that states that "a month" began three days ago (June 1, 2016).

That day was also:

• 24 Iyar, 5776 in the Jewish calendar
• 26 siyue, 4713 in the Chinese lunar calendar
• 13.0.3.8.18 in the Mayan long count calendar,
• and many others.

There is no property about those sets of days that make them "a June" -- you can't even point to temperature or the summer solstice, for it's cold and the winter solstice in the Southern Hemisphere -- so we can conclude the names of months are proper nouns, and capitalized. (They fulfill the first requirement, there are a class of "Junes:" the 2,769 of them [or so] that have occurred since the founding of Rome in 753 BCE and the commencement of the Roman calendar.)

Weekdays

Following the train of thought for months, it's easy to see that weekdays possess no quality that makes a Wednesday a Wednesday, unless you want to propose recursive qualities such as follows Tuesday, which is also an abstraction.

Another way to look at it, if you woke up on a random island somewhere, you would have no idea what day of the week it was, without someone telling you what the convention is, because that day would possess no qualities that tell you what "weekday" it is.

Again, we can point to a class of Wednesdays that are Wednesdays (by convention, or "ad hoc"), all of them that have occurred since we started calling them "Wednesday," and not Wednesdei or wōdnesdæg.