Learn Portuguese (Portugal): Pode soletrar o seu nome?

Pode is the formal way to say "you can" (matching the implied pronoun você). If you were speaking informally to a friend, you would use podes.

Why does it say o seu nome instead of just seu nome?

In European Portuguese, possessive words like seu (your) almost always need a definite article like o (the) right in front of them. You are literally saying "the your name".

Does soletrar need a connecting word like de or a before it?

No. When you use the verb poder (to be able to/can) followed by another action, the second verb stays in its raw infinitive form (soletrar) and connects directly without any extra prepositions.

If seu can mean "his" or "her", how do we know it means "your" here?

Because the verb pode is addressing the listener formally, the context makes it clear that o seu nome refers to the person you are speaking with. If there was any chance of confusion, you could say o nome dele (his name) or o nome dela (her name) instead.