I have not used thin strips of wood as a starter. You have to cut a grove for the popcicle stick, I just pour melted wax into the grove, done. The bees take to it just fine Of late I rip a 2X to 1", rip it again to 1", wiich is ripped again (twice) to a 40+ degree angle by 1/2" notch the ends, gives me a top bar with a built-in started and a 3/8" drop, used for shims.
As for glue; any carpenter glue works fine, water proof, better, fastset to speed up drying.
My 2 cents.