OK, here's two steps that should help you solve the problem.

First set the low gear limit properly. For the moment leave the cable where it is, and shift by pulling the bare wire away from the frame like a bow string. Back off the low limit until you can overshift the chain beyond low. (go easy here so you don't jam the wheel or RD). Once you can overshift, bring the limit in by degrees until you can't and stop there. Double check by backing off a half turn which should allow an overshift, then back to prevent. Now repeat shifting a number of times until you're 100% bullerptoof sure it's impossible to overshift. --- Now the Low Limit is correctly set.

[EDIT --- I think you're already covering this part OK, so it may not be relevant] Now to attaching the cable properly, and setting trim and shifting. I suspect your problem may be inability to pull up enough slack, so here's a trick. Shift to a middle gear, such that when you shift back to high without pedaling, the RD hangs inboard of the high gear limit. Now, with the lever in high position and the RD inboard of high, and with the adjuster all the way down, reattach the cable pulling all the slack out, and tighten the pinch bolt.

Having the RD inboard should help you get all the slack out, and with any luck a all, when you turn the pedals the RD should drop to high, with only vestigial slack in the wire. Now shift one click to second high. While pedaling, adjust the trim (barrel) until the RD completes the shift and fine tune the trim. Now you're almost home, shift through the gears and if 2nd high trimmed and shifted properly, all the others will too, except for some minor fine tuning.

The key is to get all the slack out, so if my trick of hanging the RD up inboard isn't enough, push it in by hand as you pull the cable down and cinch it. Also, using these methods, may cause you to over tighten the cable, so double check that you can shift to high, and if not, slowly loosen the pinch bolt until the cable slips enough to let the RD go to high.

Good luck.

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