OP I'm in a very similar situation to you at the moment - I am lucky enough to have a DP who is often around but he's very shortly got to go overseas for a few weeks and I have no real help with the DC, also with me needing a six week recovery time. I had my surgery 10 days ago and unfortunately I've had serious complications and become very unwell (hence user name). So I do think it is crucial you get that support in place now not just for a good outcome but also if you have complications and are out of action longer than planned or in hospital for a longer period. I'd also say - in hindsight had I known what I know now re my complications I wouldn't have had the surgery which was theoretically elective (for a worsening hernia so likely at some point would have become highly necessary after it got strangulated, but I could have waited until DC were older etc). If you need it you need it - and obviously complications don't happen every time so please don't let what I've said stress you out, but I'm saying it because you need to make sure you have worst case scenario cover as well.
You should ask your friends, I know it feels horrible but you will need to ask because you need that support - and if they're true friends even if they can't help much they will do something. Send out a list of things you need support with to a few close friends:
1)dog sitting for XX time and ideally being on standby to have him again if you become ill and are in hospital longer than planned - could be two separate people for these jobs
2) DD care for your hospital stay
3) Cover for DD care in case your hospital stay is extended/future stay required
4) Someone to collect you from hospital and stay with you if that's required on discharge (obviously could be you staying with them in a spare room for a night rather than them at your house).
4) Logistical help - probably the most tricky so I'd narrow this down to critical needs first ie getting DD to and from school, getting you to any urgent medical appointments etc. Again if you can get multiple supporters then you can make a little group and ask for who can do what at any given time for ad hoc things (Id have proper agreed cover for critical things like school as above).
Get food deliveries arranged so you don't have to go to the supermarket and get the easiest (whilst still nutricious) food you can. For example i have lots of wraps, salady bits like carrots peppers and cucumber that can go in wraps with cheese or make simple fajitas with, pesto pasta with peas, yoghurts for extra protein etc.
Re DD she will be OK on a reduced schedule than normal - at age 7 although she will be disappointed she is old enough to understand mummy is poorly and can't do everything she'd usually do and the two of you need to pull together. Can you get her to do little bits and bobs like bring you drinks whilst you have a couple of hours rest? If you can try to get logistical support as above for at least her most favourite activities that would be ideal but if not and if you can afford it buy in some bits you can do together that she'd enjoy (where she does most of the work so you can rest but you can advise eg craft projects etc). This should work fine for evenings and weekends short term, and hopefully you can get out and walk to the park after a couple of weeks.
Are there any school holidays during the recovery period? If so I think I'd be trying to arrange cover for her then too for some of the time to shake things up a bit, eg could she go for a sleepover at a friend's house (guess she might be too young for this though) or a few all day play dates which you could offer to reciprocate on recovery?
I think sometimes you just have to be willing to ask for help as people can be so overwhelmed with their own lives that they don't think to ask (though they should). But if it was my friend I'd be making sure I volunteered for at least some of those jobs even if it was hard - telling work I needed to slightly adjust my hours, being a bit more flexible about DC activities etc (good teachable moment about doing kind things for others), having a bit more faff than normal for a few days. Hopefully people will step up for you.
If it was me my message would basically say I'm so sorry to have to ask as I know your life is really busy and you have your own challenges but I'm really in a desperate situation and need help. [Then list your things]. Then say you will understand if they can't but any small offers of help will be worth their weight in gold to you.