Ugh. Where to begin. When I was over cub scouts some years ago, I was also flabbergasted at what was going on. I spent a lot of time researching to make sure I understood and ended up calling SL which was very insightful (I learned that the *parents* are to pay for what the boys earn (except the final one/s)). After that call, I went to the PP and explained what I learned and my ideas to clean things up (start including Faith in God, etc). After I got her blessing, I went to the bishop who also gave me his. Unfortunately, there were parents who liked the false traditions and thought the ward should still cover all the doodads so the bishop walked things back. 😕 Anyway:
Recognition - Our monthly pack mtgs were when we did Core Values/Faith in God. The parents/families were only invited (3-4 times/yr) when the boys actually accomplished things (we also had a very small group and spotty attendance). I figured there's no way boys can finish things that fast and saw no value in burdening the parents with attendance (some serve in ym/yw while the other is home with the younger kids). I don't see how you could acknowledge the girls monthly because they only meet twice and also can't learn/grasp something so quick.
Leadership - I see no problem combining. There's still plenty of time to cover the needed reqs (keep in mind I'm not familiar with the changes made in recent years) and it never hurts for the older boys to review what they supposedly already learned. Combining does have it's drawbacks though. If you're not on the same page as leaders and/or if some are flaky and drop the ball...
Budget - That is up to the PP. Here's what I suggested and it was agreed to (using general #s): $500 (total) - $250 for all of Primary (bdys, bulletin board, baptisms, nursery snacks, supplies...) - $250 split percentage-wise between scouts and AD (if have 5 boys and 5 girls - $125 ea, 5 boys and 10 girls - boys get less). Before I was put in, the previous PP gave a majority of the budget to scouts which I thought was completely wrong. We still did what we needed/wanted to do with a more balanced budget so there was no need to go back to the previous insanity.
Good luck. Personally, I'm thrilled that we're finally cutting ties with the BSA and I will never again have anything to do with that organization after seeing/learning what I have.