Might be an idea to go out for a long walk over the next couple of weekends (or whenever you have the time available) to get used to spending a lot of time on your feet. Maybe walk for 2.5 hours with some running bits thrown in. At least then it won't be a huge shock to your system on the day.
If you can run 7k, you are fit enough to walk a half marathon, so you will get round.
Get used to time on feet - it'll likely take you longer than an hour so once you have done your session, walk to make up the time on at least one of your training days.
Week 4 looks ambitious on so little grounding, so be prepared to repeat week 3. I'd be surprised if you managed to run the whole 10k on 4 weeks training (it took me 3 months to run 5k non-stop when I started running), so planned walk breaks are worth trying on the day - say run 9 mins walk 1. Worth practicing this before the day to see if it works for you.
I'm in the same boat time-wise - 2 little 'uns with "activities", working 4 days a week and the usual home stuff to fit in. If you concentrate on the 2 "important" sessions - the long slow one and the medium long one, then the others can be a quick half hour/40 mins as and when.
You know your schedule best, but remember that the loooong runs won't really kick in until 2-3 months before the marathon - maybe 9 or 10 weeks where you will be running for more than 2 hours. Only 1 other run a week needs to be longer than an hour if you are just wanting to get round.
I'm fortunate in that I have a shower at work, so can do a 35 min run at lunchtime unless I have an early afternoon meeting. My plans for next year include one midweek run of about an hour, and a long one on a Sunday morning, leaving early and running to the rugby pitches (by varying routes depending on the miles I need to get in), arriving in time to take the girls home after their training (hubby can get them up and out - I've done it all this season.....). The other runs will be on work days, and a Saturday parkrun.