Learn Hardcore French: Être invitée d’honneur la touche profondément, et Marie ressent une émotion très forte. - Being the guest of honor moves her deeply, and Marie feels a very strong emotion.

Être invitée d’honneur is an infinitive clause used as the subject of the verb touche.
Literally, it is To be guest of honour (subject) + moves her deeply (predicate), which in natural English becomes Being the guest of honour moves her deeply.
French very often uses an infinitive like this as a subject, where English would use -ing (being, doing, seeing, etc.).

Why is invitée spelled with an extra e? What is it agreeing with?

Invité / invitée here is a noun meaning guest, not the past participle invited as in English.

un invité = a male guest
une invitée = a female guest

Because the guest of honour is Marie, who is female, French uses the feminine form invitée. The ending -e marks this feminine gender.

What does la in la touche profondément refer to, and why does it come before the verb?

La is a direct object pronoun meaning her (or it, feminine). Here, it stands for Marie.
So la touche = touches her / moves her.
In French, object pronouns normally come before the conjugated verb:

Il la voit = He sees her
Être invitée d’honneur la touche = Being guest of honour moves her

Could this be rewritten in a more explicit or “straight” word order?

Yes, one natural rephrasing would be:
Le fait d’être invitée d’honneur la touche profondément, et Marie ressent une émotion très forte.
Literally: The fact of being guest of honour moves her deeply…
This makes the structure clearer for learners: Le fait d’être invitée d’honneur is visibly the subject, and la clearly refers to Marie.

Why is être in the infinitive (Être invitée d’honneur) and not something like est invitée d’honneur?

Here, être in the infinitive is acting as a noun phrase (a verbal noun): to be / being.
French uses the infinitive this way where English usually uses -ing:

Être invité ici me fait plaisir. = Being invited here pleases me.

If you said Marie est invitée d’honneur, that would be a normal finite clause with Marie as the subject: Marie is guest of honour. In the original sentence, we want the fact of being guest of honour to be the subject of touche, so the infinitive is used.

Does touche here mean physically touches? How is it being used?

Literally, toucher means to touch, but it also has a common figurative meaning: to move emotionally, to affect deeply.
In this context, la touche profondément means affects her deeply / moves her deeply emotionally, not that someone is physically touching her.
That figurative use is very common in French (and also exists in English: His words touched me.).

Why use ressent (from ressentir) instead of just sent (from sentir)?

Both sentir and ressentir can deal with feelings, but ressentir is more explicit about experiencing or feeling an emotion inside.

sentir can mean to smell, to feel (physically or vaguely emotionally).
ressentir focuses on undergoing or experiencing a feeling, often something strong or deep.

So Marie ressent une émotion très forte emphasizes that she is really experiencing a powerful emotion, which fits the context of being deeply moved.

Why is there a comma before et in …, et Marie ressent une émotion très forte?

The comma separates two related but distinct clauses:

Être invitée d’honneur la touche profondément
Marie ressent une émotion très forte

In French, it is quite normal (more so than in English) to put a comma before et when linking two full clauses, especially if you want to mark a slight pause or highlight the connection between cause (being guest of honour moves her) and result (she feels a strong emotion).

Why is it une émotion très forte and not une très forte émotion? Are both possible?

Both une émotion très forte and une très forte émotion are grammatically correct.
The default in French is noun + adjective: une émotion forte. Adding très after the noun (une émotion très forte) is very natural and quite neutral.
Une très forte émotion slightly shifts the emphasis onto très forte as a descriptive block, often giving a more expressive or literary feel. In everyday style, une émotion très forte is very common and sounds natural.

Is there any special reason for the capital Ê in Être?

Here Être is simply capitalized because it is the first word of the sentence.
The circumflex accent (ê) is part of the normal spelling of the verb être (to be).
So Être at the start of the sentence is just: capital letter + normal accented spelling.