After the verb in a negative phrase, the correct form to use more than one word or phrase, requires a “ni” at least once. “No puedes grabar mensajes de voz ni de video” is also correct.

In this one, the idea remains the same, but with fewer letters than in English.

I choose to change two elements of the last part of the sentence: the word “streaming” and the expression “en directo”. I choose to write “transmisión” instead of “streaming” because even though many Spanish speaking people know the meaning of the word streaming many others don’t. They do know the meaning of the word transmisión which has practically the same meaning, so in this case I opted for a translation that would be understandable for all the Spanish speaking people. In regards to “en directo”, I choose to put “en vivo” because in my opinion it sounds more natural and it also is a more literal translation of the expression “live”.