Basically, if you're dividing plants in fall you want to balance 1) giving them as much time as possible to make food during the summer of heat and sun and 2) gving them as much time as possible to grow new roots in the new location, before the ground freezes. I've planted stuff as late as November, but wouldn't recommend it, even for bulbs, if I had a choice (tho' I'm almost a zone 6 here at my home). Best time would be late September, I think.
You can cut back your grass either fall or spring - many people wait until spring just because the winter grass silhouette is prety. The only reason to wait on the cut-back (and the division) would be if the grass is only marginally hardy. In that case, leaving the stalks on until spring would offer some measure of protection for the growing points.
FWIW, hostas survive just fine when divided in the spring. Just do it before the leaves unfurl, or you'll go nuts. I've even divided medium-sized varieties when they were in leaf, because I didn't get to them earlier. If you really want a challenge (not that I'd recommend this activity) try moving a relatively mature specimen of one of the big blue hostas IN FULL LEAF! I did it once because I was offered a free specimen of Frances Williams, and the person was moving in the middle of summer. That plant is still the centerpiece of the bed in front of my house.

I've found it easiest to divide hosta in the spring too -- simply stab the shovel down between the points of the newly-emerging leaves.

Also, in this area bulbs can often be planted pretty late into the winter, as the ground often doesn't freeze before January. Two years in a row I've been outside in shorts and a T-shirt on New Year's Day, planting sale bulbs in 70-degree weather. They all bloomed just fine, though the stalks might have been a tad short.

Lynn

I divided and planted roses in December a few years ago, and filled the birdfeeders. No good deed goes unpunished, I promptly broke my ankle. This is year I am a good girl and am doing everything early, including treating my brugmansia for bugs before I bring them in. And they say old age doesn't make you more responsible!

I had very bad luck with the survival of divided ornamental grasses inthe fall. I had bought a bunch of grasses at an end of season sale and thought it would be smart to divide and have more (I was feeling greedy). I did and put them in plastic pots and they all died. I would think the grasses would have a better chance inthe ground and if there was suffiecent time for roots to establish.

You can move a hosta any time the spirit speaks to you. I dug some up, placed them upside down in a lasagna bed and let the roots dry for two weeks. Then I covered the foliage with more newspaper and grass clippings. Finally, I mulched the area- and had hostas come right on up!

A friend dug some up and didn't get around to re-planting them last fall. They sat on top of the ground all winter. Leafed out just fine this spring.

Spring is fine. I absolutely agree with others, anytime except the dead heat of summer is ok.

How exactly do you divide hostas? I want to relocate ones I've planted under a huge Maple tree--they just aren't thriving there. I don't know if it is too shady or if the tree is sucking up all the nutrients??? Anyway, I am going to relocate them, all but one is small, so I'd like to divide that one in half so that all the plants are the same relative size. Can I simply saw the plant in half with a garden saw?