Learn Italian: Leggo la battuta e la ricordo. - I read the line and I remember it.

You are learning that verbs like ricordare (to remember) take di when they are followed by another verb (an infinitive), such as ricordo di farlo (I remember to do it). However, when you remember a thing rather than an action, you don't use di. Here, the verb is just followed by the pronoun la (it), so no preposition is needed.

Why is la used twice in this sentence? Do they mean the same thing?

They look exactly the same but do different jobs. The first la is the feminine definite article meaning "the" in la battuta (the line). The second la is a direct object pronoun meaning "it". It replaces the feminine noun battuta and is placed directly before the verb ricordo so you don't have to repeat the noun.

Could I use the reflexive verb ricordarsi here instead?

Yes, you can! Ricordare and ricordarsi can both mean "to remember" and are largely interchangeable. If you use the reflexive version here, you would say me la ricordo. The reflexive pronoun mi (myself) changes to me because it sits right next to the direct object pronoun la (it).

I thought the word for "line" was linea or riga. What kind of line is a ?