Not unrealistic, take a look at Hal Higdons 15k/10 mile novice training plan which just happens to be 10 weeks. Plenty of other plans on the web too, have a search round and find something you like the look of and can fit your schedule.
It's possible but if you keep pushing it like that you're likely to end up injured and unable to run. It's great that you want to continue running but the body needs time to adapt.
If you go for it good luck, add a mile a week to your long run with a couple of cut back weeks and 2 weeks taper should get you through, but my recommendation would be to give yourself more time to build up to half marathon distance.