Pourriez-vous me donner un gâteau?

Always infinitive with the translation of the modal can.

Can you… Could you.. Will you…
All give infinitive when you translate in French.

To the first question, you must use

Pourriez-vous me donner un gâteau ? (infinitive form)

And for the last

Pourriez-vous lui donner un gâteau ? (personal pronoun, 3rd person)

Your se was an error, because you used the reflexive form, which is not the case here (You give him a cake)

I don’t know what exactly is unclear to you, could you precise ?

A verb form that varies depending on the person always has a subject indicating the person. A verb that is a complement of another verb lacks a subject and can only be in an impersonal mood: infinitive or participle.

For example, in a sentence like “Pourriez-vous me donner un gâteau?”, there are two verbs: pourriez and donner. The verb pourriez is the main verb of the sentence; it is in the conditional mood and its subject is vous. The verb donner is a complement of pourriez; it has no grammatical subject, and it is in the infinitive mood.

Apart from the passive voice and from past tenses, most of the time, when a verb is a complement of another verb, the complement is an infinitive. The alternative is that the complement is a full subordinate clause with a subject and a verb in a personal mood (indicative, subjunctive or conditional). Usually, when the semantic subject of the main verb and the complement verb are the same, we use an infinitive, and not a full subordinate clause.