Inga, relax! I've just planted my tank, but I am quite a good gardener on land, so, even though I am anticipating failure, I'm not stressing out about it. Plants just need what they need and if you give them that, they'll be fine. If they don't, they end up in the compost.
I'm having exactly the same issue as you with the plants coming loose. The fish seem to dislodge the plants. I went out and bought another bag of gravel and, after pushing the plants into the existing gravel, I gave each an extra handful around the bottom. It seems to have helped.
As for getting the plants in the gravel, I found that I needed to use my hand, but my aquarium cleaning "sponge with handle" has a fork on one end for planting plants. It can be a bit fiddly, but I can get the plants back down sometimes using it to push and wriggle them into the gravel. If your tank is too deep, you might find something similar useful (although it's not as easy as shoving them in with your hand).
As already mentioned, pennywort can float or be planted. I've had a couple of stems floating around for months and have only just planted it in the substrate (I did plant it originally but, like yours, it became dislodged. I just left it floating and it is still alive - not thriving (presumably because I wasn't adding any fertilizer), but alive nonetheless.)
As for dividing your plant, just go for it
. You won't break the end off if you are pulling it down and apart, you'll end up with separate pieces with the ends intact. I don't know the plant, but it may not even matter if you don't get any roots on each piece, it might still be able to be planted and it'll grow roots (someone else will have to tell you if it matters). Don't worry about wrecking anything, you won't. Edit: Looks like you've already done it! Forget this bit.
(Maybe we need a support group for us nervous new plant owners?