Learn Spanish (Latin America): El doctor abre la puerta cuidadosamente.

It comes from the adjective cuidadoso (careful). To make it an adverb meaning carefully, you first change the adjective to its feminine form, cuidadosa, and then attach -mente to the end.

Could I put cuidadosamente before the verb, for example: El doctor cuidadosamente abre la puerta?

Yes, you can. In Spanish, adverbs ending in -mente usually go immediately after the verb or at the very end of the sentence. Putting it before the verb is also correct and just adds a little extra emphasis to the fact that he was being careful.

Is cuidadosamente related to the phrase tener cuidado?

Exactly! They both share the root noun cuidado (care). While tener cuidado is a verb phrase meaning to be careful (literally to have care), cuidadosamente is a single word used to describe exactly how an action is being done.

Why is the verb abre instead of abra?

Abre is the regular present tense of the verb abrir (to open) for he/she/it. You might be remembering abra, which is the formal command form used to tell someone to open something. Since the doctor is just doing a normal action here, we use the regular present tense.