EX13-053 Thundermon

It refers to a card that contains the specified text or icon in its name, traits, effects, inherited effects, (Rule), digivolve requirements, DNA Digivolve, DigiXros requirements, Burst Digivolve, App Fusion, Link, or Assembly requirements.For example, "a card with [Knightmon] in its text" would include cards with the name [DarkKnightmon] and cards with the text [Knightmon] in their effects.

Yes.Additionally, even if you returned this very card from the trash to the deck, you can still process the rest of this [On Play] [On Deletion] effect until the end.

No, it is not treated as [Mamemon].The name of this card is [Thundermon], but the (Rule) references it as a card that has [Mamemon] in its name.