Houd mijn tas even vast terwijl ik de deur open.

Because the full verb is vasthouden, which is a separable verb. In many sentence types, Dutch splits separable verbs into two parts:

• infinitive: vasthouden
• main clause: Ik houd mijn tas vast
• imperative: Houd mijn tas vast
• subordinate clause: ... omdat ik mijn tas vasthoud

So houd ... vast belongs together as one verb meaning hold on to / keep hold of.

Here even does not literally mean even in the English sense. In spoken Dutch, even often softens a request and can mean something like for a moment, just, or simply make the sentence sound more polite.

So Houd mijn tas even vast feels more natural and less abrupt than just Houd mijn tas vast.

Because terwijl introduces a subordinate clause, and in Dutch subordinate clauses the conjugated verb usually goes to the end. That is why you get:

terwijl ik de deur open

In a main clause, the order would be:

Ik open de deur.

So the difference is mainly about clause type and Dutch word order.

It is a verb here. Specifically, it is the first person singular present tense of openen:

• ik open = I open
• jij opent = you open
• hij opent = he opens

Because it is in a subordinate clause, it appears at the end: terwijl ik de deur open.

Yes. Dutch has several natural ways to express open the door:

• de deur openen
• de deur opendoen
• de deur openmaken

All are understandable. Openen can sound a bit more neutral or slightly more formal, while opendoen and openmaken are very common in everyday speech.

Dutch often makes requests polite without using a direct equivalent of please every time. Words like even, tone of voice, and context do a lot of that work.

If you want, you can add alsjeblieft:

Houd mijn tas even vast, alsjeblieft.

But it is not necessary for the sentence to sound polite.

Because deur is a de-word in Dutch. The article has to be learned with the noun:

• de deur
• de tas

Unfortunately, there is not always a simple rule for whether a noun takes de or het, so learners usually have to memorize it.

In Dutch, final voiced consonants are often devoiced in pronunciation. So the final d in houd is pronounced more like a t sound.

That means houd sounds roughly like hout at the end, even though it is spelled with d because of the verb houden.

Yes, you will often hear and see hou as well. Houd is the fuller spelling, but hou is very common in everyday Dutch.

So both may occur, and learners should be prepared to recognize both forms.

Because terwijl specifically expresses that two actions happen at the same time: you hold the bag, and I open the door.

Als usually means if or when, so it would not be the best choice here. Terwijl is the natural word for while in this sentence.