Je me mange une pomme is a case of "autobenefactive" form. It is considered informal outside regions where it is very common (Southern France).

Je me cherche du travail/un emploi is more usual because while you can’t eat an apple for someone else, it is still possible to look for a job for someone else (e.g. Je cherche du travail pour mon fils.)

However, if you just say elle cherche un emploi without more precision, everyone will assume she is doing it for herself so elle se cherche un emploi can be considered slightly redundant but nevertheless common, especially in Canadian French. Regardless of the grammatical difference, there is no difference in meaning between je cherche du travail et je me cherche du travail.

Agripper is transitive direct: J’agrippe son bras.

When used pronominally, the direct object is the reflexive pronoun me so son bras becomes a complément d’objet second and requires the preposition à: Je m’agrippe à son bras, i.e. literally "J’agrippe moi à son bras".

On the other hand, while manger is also transitive direct, me is not the direct object because I do not eat myself here. It is still the apple that is eaten so the apple stays the direct object: Je me mange une pomme, i.e. "Je mange une pomme (pour/à) moi".