I forget if there's an equivalent for the first one, but from psql there is a translation of mysql's "DESC table" as "\d table", and the rest are:
\dt
\l
SELECT * FROM pg_stat_activity;
\dt
\l
SELECT * FROM pg_stat_activity;
Not really; PostgreSQL doesn't store the original query, so you'll need to re-create it from pg_class, pg_attribute, and all of that (which is really what \d and such in psql do). The easiest way is probably pg_dump, but it's best to just get used to \-commands because it's really just the same thing.