St. Gabriel grub control - Purchased 20 lb. bag

In terms of grubworm control, 'milky spore' products are bacterium meant to create a long-term soil-borne epidemics amoung grubs in the soil. Milky spore very likely will not kill existing populations that're eating at turf roots & thus, doing damage.
But milky spore is an environmentally-conscious product, and certainly a wise long-term investment for those who've put a lot of investment in their lawns.

In leiu of milky spore, I recommend any product containing Dylox for active infestations.
Milky spore may be applied in that same season in an attempt to start that longer-term, human & pet-safe grumworm contagion.

It is perfectly normal to see small numbers of grubs in the soil, in both turf & landscape. In small numbers, grubs are generally harmless. The concern mounts when their numbers are higher.

This is how to check for actively feeding grubs:
In turf areas that are turning brown, get on your hands & knees and take the turf into both hands, and pull.
If the turf readily comes up as if it's carpet, you'll likely see the "lawn shrimp" (haha) underneath.
In a 12" x 12" section, if you see more than 2 grubs, it's time to take action.

Answers:
1) No. After applying Dylox & watering in per instructions, wait approx 1 week to 10 days, then peel back some of those sod areas again.
If it's working, the grubs will have slowly turned from an ugly grey to a light beige, and shrunken somewhat.

2) No. Moisture on grass at time of application doesn't matter. What matters is that you irrigate thoroughly AFTER application.

3) Actually, it is to your advantage if adjacent soil areas aren't treated with milky spore. If ALL grubs are killed, then there will be no carriers to continue the bacterial contagion. Even if your neighbors are grub-free, keep in mind that Japanese, June beetles, masked chafer, etc will drop eggs into the turf season after season. (There's nothing you can do about that.) And these eggs will eventually metamophecize into grubs. If you have milky spore in the soil, they shouldn't mature to the point to which they'll feed.

4) Products containing Dylox are labelled to kill grubs of all species of beetle. At about the time you're seeing June beetles swarming your back porch light, they're likely laying eggs in the turf. Most species of grubworm potentially can do damage 2X year:
JUST before the adults emerge, and about a month to six weeks after the maximum height of adult population.
Here in s. Ohio, late September thru first ground freeze tends to be historically the worst peroid for grubs. But some years, mid to late spring damage can be found, too.