Learn Hardcore Japanese: kotira no doa kara haitte kudasai. kaigisitu ha migi desu. - Please enter through this door. The meeting room is on the right.

こちら is a polite demonstrative meaning “this way/this side/this place.” こちらのドア literally means “the door over here/on this side,” and sounds more courteous than このドア.
この modifies a noun directly (“this door”), and is fine in neutral speech: このドアから入ってください.
ここ is an adverb (“here”), so you can’t put it directly before ドア. ここのドア is grammatical (“the door here”), but in guidance/signage こちらのドア feels more natural and polite.

What does the の do in こちらのドア?

の turns こちら (“this side/this place”) into a modifier for ドア, making “this side’s door” → “this door here.” Functionally it’s like using この, but こちらの is a touch more formal/polite.

Why use から after ドア? Could I use を, に, or で instead?

With 入る (“to enter”), から marks the point or route of entry: ドアから入る = “enter through/from the door.”
に marks the destination: 部屋に入る = “enter the room.”
を is generally not used with 入る in modern speech (門を入る exists as an old/literary pattern). If you want を for a route, use a different verb: ドアを通ってください (“please go through the door”).
で marks means/instrument, so ドアで入る is not natural.

Is 入ってください the best way to say “please enter”? What about お入りください or くださいませ?

入ってください is a polite, everyday request.
お入りください uses the honorific お + verb stem + ください; it’s common on signs and sounds more refined.
お入りくださいませ is even more formal/customer-service-like.
Spelling note: for requests, ください is usually written in hiragana (not 下さい).

Why is there a small っ in 入って (はいって)? How is it formed and pronounced?

入る (はいる) is a godan/u-verb ending in る. Its て-form changes る → って: 入って (pronounced はいって, “haitte”). Brief rule:

う/つ/る → って (買う→買って, 立つ→立って, 取る→取って)
む/ぶ/ぬ → んで (読む→読んで)
く → いて (書く→書いて) except 行く→行って
ぐ → いで (泳ぐ→泳いで)
す → して (話す→話して)

Is 会議室は右です natural, or should it be 右にあります or 右側です?

All are fine; nuance differs:

会議室は右です is a concise, conversational way to say “It’s to/on the right” (elliptical for “右のほうです”).
会議室は右にあります explicitly marks location/existence; a bit more precise.
会議室は右側です emphasizes “the right-hand side.” Any would be understood.

What is the role of は in 会議室は右です?

は marks the topic: “As for the meeting room, [it] is on the right.” It sets 会議室 as the thing being discussed. Using が here (会議室が右です) would sound like identifying “the thing that is right is the meeting room,” which is odd unless contrasting/answering a very specific question.

How can I say “on your right” more explicitly?

Use:

右手にあります (“it’s on your right-hand side”)
右側にあります / 右側です (“on the right side”)
右のほうです (“toward the right”) Very polite: 右手にございます.

Can I omit ドア and just say こちらから入ってください?

Yes. こちらから入ってください (“Please enter from here/this way”) is natural and common. If there are multiple entrances and you want to be explicit, こちらのドアから is clearer.

If the door is not near me, should I use そちら or あちら instead of こちら?

Yes, use the ko–so–a series:

こちら = near the speaker (“this way/this door”)
そちら = near the listener (“that way/that door” by you)
あちら = far from both (“that way/that door over there”) So: あちらのドアから入ってください if it’s away from both of us.

Is ドア the only word for “door”? What about 扉 or 戸?

ドア is the general, everyday loanword for doors (especially hinged).
扉 (とびら) is a slightly more formal/written word; often seen on signs or manuals.
戸 (と) often refers to traditional sliding doors or doors in a Japanese-style setting. In an office/building, ドア or 扉 are both fine.

What’s the difference between 入ってください and 入れてください?

入ってください = “Please enter” (you perform the action of entering).
入れてください = “Please let me/us in” (you’re asking someone to allow/enable the entering; transitive verb 入れる “to put/let in”).