Does "Mountainwalk" imply "Snow-Covered Mountainwalk"?
Yes.

Since Snow-covered is a land supertype, and mountain is a subtype, a snow-covered mountain is also a mountain and thus plain vanilla mountainwalk applies. The land subtypes are specified in rule 204.3h


  204.3h Lands have their own unique set of subtypes; these subtypes are called land types. The land types are Desert, Forest, Island, Lair, Locus, Mine, Mountain, Plains, Power-Plant, Swamp, Tower, and Urza's.
            Of that list, Forest, Island, Mountain, Plains, and Swamp are the basic land types. See rule 305.6.


However, you are correct about the converse. A creature with Snow-covered mountainwalk would be able to be blocked if you had a mountain but no snow-covered mountains.