This perception works both ways - Russian sounds/feels like an archaic version of Czech.
And the feeling is similar with other Slavic languages -- simply because each Slavic languages preserved different features from the old Proto-Slavic language. Czech is actually objectively a bit more progressive - probably because it evolved side by side with German.
Not really... With most Czech verbs, there is no distinction between English simple and progressive tenses, but verbs of motion (like go), which include "nést" are kinda special, they come in pairs - one means an immediate, one-time action, the other means a repeated or habitual action. Here it's "nést" - one-time, and "nosit" - repeated. So "I carry pears" typically translates to "Nosím hrušky."