"<x> calculus" without an article is an activity, whereas "the <x> calculus" is a concept. That's the way I've always parsed it internally. For example, you would never say to someone "Do you know about SKI Calculus?" because it's not something people do. You would say "Do you know about the SKI Calculus", referring to the abstract notion that someone constructed. We often use the phrase "calculus" for the calculus of derivatives and the calculus of integrals because it's something people do as an activity very often. Same with the lambda calculus.

That may work for your internal parsing, but it's not correct with respect to any standard of English grammar. Consider that any activity can also be a topic of knowledge, and it doesn't change article usage: "do you know about football?"

"The", the definite article, has a couple of purposes in English, but by far the most common is to refer to an individual member of a set of nouns. The decision to use it or not in this case depends on how the the words "lambda calculus" are interpreted:

1. "Lambda calculus" is a compound noun that describes a unique or uncountable object or concept. In this case, "lambda calculus" is in effect the only member of the set to which it belongs, and so "the" is not used. Compare with any other conceptual noun, e.g. "mathematics".

2. "Lambda calculus" is a compound noun, but is not unique - that is, there are multiple things that could be referred to as "lambda calculus". In this case, "the" is used to select a particular lambda calculus. An extended phrase might be "the lambda calculus introduced by Alonzo Church, as opposed to the lambda calculus introduced by Joe Shmoe." Note also that in general parlance, "the" can be used alone to refer to the most common one of something, as if that member of the set were always considered an antecedent to current conversation. For example, in certain circles in California, "the industry" refers to the entertainment industry. Because it's so common to refer to that particular member of the set of all industries, "the" on its own is sufficient to select it.

3. "Lambda calculus" is a generic noun (calculus) with a modifier (lambda) that makes it unique. In this case, similar to (2), "the" must be used, as it along with "lambda" serves to select the particular calculus being referred to. Compare with "the Nile River" - "Nile River" is unique, but since "Nile" is a modifier for a generic noun the article is required. (Following the final point from (2), saying simply "the calculus" would under this interpretation normally refer to the calculus of derivatives and integrals, since that's by far the most common.)

4. Same as (3), but "calculus" is not countable in the same way, and so it does not make sense to refer to the set of all things known as "calculus". In this case, "lambda calculus" refers to a portion of the singular concept "calculus" rather than a member of a set, and so "the" is not used.

I think the most common intentions are (3) and (4) for using or not using "the", respectively. (4) is more common among non-mathematicians for whom "calculus" is a singular subject in school, while (3) is more common among people who are aware that "calculus" is a general term for "form of calculation", and who might refer to the particular "calculus" taught in high school as "the calculus of derivatives and integrals". I'm normally partial to (1), myself (i.e., "lambda calculus" and "integral calculus" are each topics of knowledge, grammatically identical to "mathematics").

Good question. I think it's just an oddity/ambiguity of English (and other languages as well, I'm sure). I know that calculus (with derivatives and integrals) has been called "the calculus" in the past. I mostly just say "the lambda calculus" because that's what I've heard.

Another way that it seems consistent in my mind is that there are many calculi (systems for computing things) out there, and the lambda calculus is one particular calculus that uses lambda expressions as the building block. The pi calculus is another example. The "the" feels a little more appropriate when "lambda" is seen as a modifier specifying which calculus I'm referring to.

In other cases, I'll drop the "the", though. For example, I might say "There are three types of lambda calculus expressions.".

Your final case isn't specific to "lambda calculus". In that sentence, you're using "lambda calculus" as a modifier for the general term "expressions", rather than referring to it on its own. Consider that you would say "There are three types of ceiling lights", but you would still say "the ceiling".