No problem. A lugged sole is one that is knobby, for increased traction. If you have mountain bike shoes, chances are that they have lugged soles.

A hard-tail mountain bike is one that has no rear-end shock absorber. Many mountain bikes these days have a rear shock, which is nice, but it makes the bike heavier, and isn't necessary for 'cross. It sounds like your mountain bike is heavy, and you therefore you probably wouldn't want to haul it on a 'cross course.