Task Resolution
I'm not a fan of the "roll under your ability score on a d20 method -- it's too broad and doesn't allow for difficulty, IMO. On the other hand, I don't think a d20-ish solution would work well with OD&D, given the way OD&D does ability modifiers, etc. I'd go with one of the following, depending on the situation:
1. Roll under ability using Xd6. It gives you a bell curve, and you can shift the difficulty higher or lower by modifying the number of d6s that are rolled (i.e. 2d6 is easy, 3d6 is average, 4d6 is hard, et cetera). Works well if level/skill isn't a big factor, but gets weird if level/skill should be a factor.
2. DM assigns a % chance based on the circumstances and the PC performing the action, and the player rolls d100.
Acting Order in Combat
I see you're using the Holmes system. I'd not roll monster Dex unless there is some question. That is, I'd just assign Dex to something like a panther or a giant slug, and only roll if it wasn't obvious (e.g. an orc). (Also, I suggest staying away from the "light weapons hit twice, two-handed weapons go every other round" stuff from Holmes, just in case you're considering that, too.)
I usually do this:
1. If it's obvious an action or PC should go first, he goes first, et cetera.
2. Then by Dex.
3. For ties/very close, resolve with an initiative check.
(That is, common sense->Dex->die roll)
Critical Hits and Misses
For critical hits, I allow maximum damage on a natural 20, with no need to roll for damage. I think this models "the best hit possible" without unbalancing anything. Plus, it's not any extra die rolls.
For natural 1s, I ask for an ability check or saving throw appropriate to the situation; failure means something bad happened. I usually just rule on that (e.g. drop your weapon, shoot your friend, fall down, et cetera).