Learn Hardcore Spanish (Spain): Agradezco haber ofrecido mi ayuda a mi amiga. - I am grateful to have offered my help to my friend.

agradezco can only be yo (I), so saying yo agradezco is not necessary.
You would add yo only for emphasis or contrast, e.g. Yo agradezco haber ofrecido mi ayuda, pero otros no.

What tense is agradezco, and what does it express here?

agradezco is present indicative (first person singular of agradecer).

Here it expresses a present feeling (I am now grateful / I now appreciate) about a past action described by haber ofrecido mi ayuda (having offered my help). So structurally:

Present feeling: agradezco
About a prior action: haber ofrecido mi ayuda a mi amiga

What exactly is haber ofrecido, grammatically speaking?

haber ofrecido is the perfect infinitive:

haber = infinitive of haber (to have, as an auxiliary)
ofrecido = past participle of ofrecer

The perfect infinitive is used to talk about an action that happens before the action of the main verb, when both share the same subject.

So:
agradezco (now) → haber ofrecido (earlier action I’m thankful for)

Why do we say haber ofrecido and not he ofrecido?

Because in this structure the main verb (agradezco) is already conjugated. The second verb must be in an infinitive form, not a finite tense.

he ofrecido is a full conjugation (present perfect: “I have offered”).
In Agradezco haber ofrecido…, haber ofrecido acts like a noun phrase: “having offered…”.

So you can’t say:

✗ Agradezco he ofrecido mi ayuda… (two finite verbs together)
You must use the infinitive:
✓ Agradezco haber ofrecido mi ayuda…

Could we say Agradezco ofrecer mi ayuda a mi amiga instead of haber ofrecido? What’s the difference?

You can say Agradezco ofrecer mi ayuda a mi amiga, but it sounds more like:

“I’m grateful to offer my help to my friend (in general / as a possibility).”

Using haber ofrecido makes it clear that the offering is a completed past action:

Agradezco haber ofrecido mi ayuda… = “I’m grateful for having offered my help (on that occasion).”

So:

ofrecer (simple infinitive) → more general, simultaneous, or habitual.
haber ofrecido (perfect infinitive) → clearly prior, completed action you’re now grateful for.

For a specific past event, haber ofrecido is the natural choice.

Who is the subject of haber ofrecido mi ayuda?

The subject is the same as the subject of the main verb, i.e. yo (implied).

So the meaning is:

Yo agradezco [yo haber ofrecido mi ayuda a mi amiga].

Spanish doesn’t repeat yo before the infinitive, but logically it’s the same person.

Why doesn’t ofrecido agree in gender or number with ayuda?

With the auxiliary haber, the past participle never changes for gender or number. It’s always invariable:

he ofrecido, has ofrecido, han ofrecido, haber ofrecido → ofrecido stays the same.

Agreement (changing ending for gender/number) only happens when the participle is used as an adjective, e.g.:

la ayuda ofrecida, las ayudas ofrecidas

In the sentence haber ofrecido mi ayuda, ofrecido is part of a verbal form, not an adjective, so it doesn’t agree with ayuda.

Why is it mi ayuda and not just ayuda or la ayuda?

All three are possible in different contexts, but they don’t feel the same:

mi ayuda – clearly “my help,” emphasizing that I am the one who helped.
ayuda (without article/possessive) – more general, sometimes sounds like “help” in the abstract.
la ayuda – “the help,” referring to some specific help already known in context.

In this sentence, mi ayuda fits nicely because you’re talking about your own offer of help:
Agradezco haber ofrecido mi ayuda a mi amiga.

What is the function of a in a mi amiga? Why not just mi amiga?

The a here is the normal preposition used with verbs like ofrecer when you indicate to whom something is offered:

ofrecer algo a alguien → “to offer something to someone”

So:

ofrecer mi ayuda a mi amiga = “to offer my help to my friend”

The a is not optional; without it, mi amiga would look like a direct object, but mi ayuda is already the direct object. a mi amiga is the indirect object, marked with a.

What’s the difference between a mi amiga and para mi amiga here?

a mi amiga focuses on the person who receives the action (the one you offer your help to). It’s the standard pattern with ofrecer:
ofrecer algo a alguien
para mi amiga focuses on the purpose or intended beneficiary: “for my friend / for my friend’s benefit.”

In this sentence, the natural, idiomatic option with ofrecer is a mi amiga, not para mi amiga.

Can a mi amiga be replaced by a pronoun? How would that look?

Yes. You can replace a mi amiga with an indirect object pronoun:

Agradezco haberle ofrecido mi ayuda.
(Here le = a mi amiga.)

If you also replace mi ayuda with a pronoun, you get:

Agradezco habérsela ofrecido.
(se = le changed before la, and la = mi ayuda.)

Note that with infinitives, object pronouns normally attach to the end:

haberle ofrecido
habérsela ofrecido

If I want to say “I appreciate that my friend offered me help,” how would I change the sentence?

For “I appreciate that my friend offered me help,” your friend becomes the subject of the offering, not you. You need a subordinate clause with subjunctive:

Agradezco que mi amiga me haya ofrecido su ayuda.

Structure:

Agradezco que… (I appreciate that…)
mi amiga = subject of the subordinate clause
me haya ofrecido = present perfect subjunctive
su ayuda = her help

Is Agradezco haber ofrecido mi ayuda a mi amiga something people in Spain would naturally say, or is another verb more common?

The sentence is grammatically correct and understandable in Spain, but in everyday speech many people would more naturally use me alegro de instead of agradezco for this idea:

Me alegro de haber ofrecido mi ayuda a mi amiga.
(“I’m glad I offered my help to my friend.”)

Agradezco haber ofrecido… is a bit more formal or introspective, as if you are consciously appreciating your own past decision. It’s not wrong, just stylistically marked compared with the more common Me alegro de….