One piece of information re your Abelia which would be very useful for ID is whether it's evergreen or not - if it is evergreen, it's mostly likely A. grandiflora, and the best way to propagate that particular one is hardwood cuttings in autumn, left outside to root in a cold frame. If you have a warm, sunny spot in the garden where you could take some more, longer, hardwood cuttings, insert them in the ground and cover with a cloche or similar, anchored down, these may root as well over winter and next spring. If its not evergreen, then what you've done already should suffice.
Suggest you clip off the flowering part immediately - it does not signify the production of roots on the cutting, the 'information' in that particular part of the plant was already coded for flowering when you took the cutting, so those particular cells are flowering regardless. They will detract from the cutting's ability to produce roots, so cut them off immediately.
Your part of France, I believe, is Zone 8, the same as the UK. The best way to overwinter your cuttings is in a cold frame placed in a warm part of the garden, closed during severe weather, up against the house wall if possible for extra warmth/shelter. Whether you want to buy a moveable cold frame with a base is your decision, but perhaps you could think of something you can cobble together to create a similar situation. I've used cardboard boxes with the front cut down to 1 inch and the sides angled down to meet the front, then covered the whole lot in plastic, but that's really only good for a few weeks in summer.
I wouldn't recommend moving the cuttings out of their pot at this stage, unless you know they have good roots already. What you can do is put the pot inside a larger pot and pack the gap with compost. It's probably wise to wrap the outer pot in fleece, or surround it with straw or any kind of insulating material to try to prevent the pot freezing - if you have very severe weather conditions, with temperatures falling below -3deg C during the day and staying there for a week, it is at risk of freezing, and you may need to move it temporarily into a cool or cold windowsill until the weather eases back a bit, when it can be replaced outside. A covering of snow on your 'cold frame' is a good thing - it insulates well and means the pot it is a little less likely to freeze. Raising the cold frame off the ground would also help.