As for the rules : " A mini can move into base contact with an enemy mini only if the unit performing the move has a melee weapon and is entering into a melee with that enemy unit"

Can a unit move into melee if it has no actions left to start a melee attack this turn? For example , can a unit move twice in a turn and enter in melee, waititng for next turn to do an attack in melee?

Yes. It gets even better if the unit they enter melee with another troop unit that has activated already as there is next to nothing your opponent can do about it this turn. (Except counter charge and force push)