Carbs need to be synchronised.
The stand that lifts the rear wheel off the ground is the centre stand. The stand that the bike leans over on is the side stand or kick stand.
Most Kawasakis have a positive neutral finder that prevents 2nd gear from engaging if the bike isn't stationary. That might be causing your 2nd gear issues. With the box stripped you should be able to spin the shafts and operate the shift mechanism and see what's going on.