How can I adjust index shifter so it doesn't skip the second chainring?
You went at it backwards.  First thing to do (after backing off the limit adjustments a bit) is to adjust the barrel adjuster so that you can hit middle gear, then adjust the upper and lower limits so that you hit the outer two without going too far.  Then you can "tweak" the barrel adjuster slightly so that you minimize chain rub at the rear without upsetting clean shifting to the middle gear at the front (then "tweak" the limits again).