Learn Russian: Мой папа говорит медленно.

The dictionary form of this verb is говорить (to speak). Because it ends in -ить, it belongs to the second group of Russian verbs (the second conjugation). For a "he", "she", or "it" subject like папа, we drop the -ить and add the ending -ит. Verbs from the first group usually get an -ет ending here instead.

The word папа ends in -а, which is usually a feminine ending. Why do we use the masculine word мой (my)?

You're right that most Russian nouns ending in -а are feminine. However, папа refers to a male person, so Russian always treats it as a masculine noun. That is why it must pair with the masculine possessive pronoun мой rather than the feminine моя.

How can I tell that медленно is an adverb meaning "slowly" rather than the adjective "slow"?

In Russian, many adverbs (words that describe how an action is done) are formed by taking an adjective and changing the ending to -о. The adjective for "slow" is медленный, but since we are describing how the dad speaks, we use the -о ending to make it медленно (slowly).

Does this sentence mean "My dad speaks slowly" in general, or "My dad is speaking slowly" right now?

It can mean both! Russian only has one present tense. Мой папа говорит translates to both "My dad speaks" as a general habit and "My dad is speaking" right at this exact moment. You just have to guess which one based on the context.