Learn Hardcore Turkish: Ben kutuyu kapıcıya taşıtmak istiyorum. - I want to have the doorman carry the box.

The whole phrase kutuyu kapıcıya taşıtmak functions as the thing that is wanted.
istiyorum is the only finite verb (conjugated for person and tense).

You wouldn’t say “Ben kutuyu kapıcıya taşıtıyorum istiyorum.” Only istiyorum is conjugated; taşıtmak stays in infinitive form.

What does the -t- in taşıtmak do? Is it always “make someone do something”?

The -t- is a causative suffix. It adds the meaning:

“cause / make / have someone do [the action].”

So:

yazmak = to write
yazdırmak = to make/have someone write
okumak = to read
okutmak = to make/have someone read
taşımak = to carry
taşıtmak = to make/have someone carry

In context, the English nuance can vary:

make (force)
have / get (ask, arrange, pay)
let (allow, in some other verbs)

In this sentence, the most natural reading is something like:

I want to have the porter carry the box
(i.e. I want to arrange for the porter to do it, not me).

Why is kapıcı in the dative case (kapıcıya)?

In many causative constructions, the person who is caused to do the action (the “doer” you control) is put in the dative case (-a / -e).

General pattern:

[Causer in nominative] + [Causee in dative] + [Object in accusative] + [Causative verb]

Examples:

Ali çocuklara masal okuttu.
Ali made the children read a story / had the children read a story.
Patron işçilere fazla mesai yaptırdı.
The boss made the workers do overtime.

Here:

Ben = causer
kapıcıya = causee (the person who will actually carry)
kutuyu = direct object
taşıtmak = causative verb (have someone carry)

So kapıcıya is not “to the porter” as a destination here; it is “to the porter” in the sense “to the porter to do the job” → “have the porter do it.”

Who is actually carrying the box in this sentence: me or the porter?

The porter is actually carrying the box.

Roles:

Ben = the person who wants / arranges / causes the action
kapıcıya = the person who performs the carrying
kutuyu = the box that gets carried

So the meaning structure is:

I want [the porter] to carry the box
= I want to have the porter carry the box.

If you yourself were carrying the box, you would say:

Ben kutuyu taşımak istiyorum.
I want to carry the box.

Why is it kutuyu and not just kutu?

Kutuyu is kutu + -(y)ı (accusative case).

In Turkish, a definite, specific direct object is normally marked with the accusative:

Kutu taşıyorum. = I am carrying a box. (indefinite)
Kutuyu taşıyorum. = I am carrying the box. (definite, specific)

In the causative:

Kapıcıya kutu taşıtmak istiyorum.
I want to have the porter carry a box. (not specific)
Kapıcıya kutuyu taşıtmak istiyorum.
I want to have the porter carry the box. (a particular box)

So kutuyu shows that we’re talking about a specific box known from the context.

Can Ben be left out? Is it necessary here?

It’s not necessary; it can be omitted.

From the verb ending -um in istiyorum, we already know the subject is “I”.

So:

Ben kutuyu kapıcıya taşıtmak istiyorum.
Kutuyu kapıcıya taşıtmak istiyorum.

Both mean “I want to have the porter carry the box.”

Using Ben typically adds emphasis or contrast:

Ben (not someone else) want to have the porter carry the box.

Can I change the word order? For example, say Ben kapıcıya kutuyu taşıtmak istiyorum?

Yes. Turkish word order is flexible, as long as the verb stays at the end (in neutral sentences).

All of these are grammatically fine:

Ben kutuyu kapıcıya taşıtmak istiyorum.
Ben kapıcıya kutuyu taşıtmak istiyorum.
Kutuyu kapıcıya taşıtmak istiyorum.
Kapıcıya kutuyu taşıtmak istiyorum.

The basic meaning doesn’t change, but the focus / emphasis can shift. Roughly:

Phrase closest to the verb often carries slightly more emphasis.
If you say Ben KUTUYU kapıcıya taşıtmak istiyorum, stress is on kutuyu (it’s that box).
If you say Ben kapıcıya KUTUYU taşıtmak istiyorum, stress is still on kutuyu, just with a different rhythm.

For most learners, it’s enough to know that swapping kutuyu and kapıcıya is OK and doesn’t change the core meaning.

How would I say the same idea without using the causative taşıtmak?

You can say it with a subordinate clause using -mesini istemek:

Kapıcının kutuyu taşımasını istiyorum.
I want the porter to carry the box.

Compare:

Ben kutuyu kapıcıya taşıtmak istiyorum.
- Literally: I want to have the porter carry the box.
- Emphasizes that I want to arrange / cause this to happen (I might be in charge, paying him, etc.).
Kapıcının kutuyu taşımasını istiyorum.
- Literally: I want that the porter carry the box.
- More neutral: I simply want the porter to carry it (no built-in causative nuance).

Both can often translate as “I want the porter to carry the box”, but taşıtmak explicitly encodes “making / having someone do something” as part of the verb itself.

How would I say “I made the porter carry the box” (past tense) using this pattern?

Use the past tense of the causative verb:

Ben kutuyu kapıcıya taşıttım.
I made / had the porter carry the box.

Breakdown:

taşı- = carry
-t- = causative
-tı- = past tense (-dı / -di / -du / -dü → here becomes -tı by vowel harmony)
-m = 1st person singular

Spelled together: taşıttım (the two t’s meet: causative -t- + past -tı-).

So:

taşıtmak istiyorum = I want to have (someone) carry (it)
taşıttım = I had / made (someone) carry (it)

Why is it taşıtmak istiyorum and not just taşıyorum?

Because the meanings are different:

taşıyorum = I am carrying (now / currently).
taşımak istiyorum = I want to carry (myself).
taşıtmak istiyorum = I want to have someone carry (it).

In the original sentence:

Ben kutuyu kapıcıya taşıtmak istiyorum.

you are not saying you are carrying anything right now. You are stating a desire / intention to arrange for the porter to carry the box. Hence:

istiyorum (I want)
plus infinitive taşıtmak (to have [someone] carry).