She will be absolutely fine. It's actually the best age to travel because they aren't yet mobile and still sleep a lot - and there will be two of you, so you can at least free up your hands for things like using the toilet, etc. I took my youngest to Spain for a week on my own at 8 months and it was also fine - but more work because it was just me.
No suggestions for a pram - I just used my same one as I used at home, got a buggy bag for it, and it got checked on the plane. I'd just look for a cheap secondhand one off Facebook marketplace, but one that can still lie flat or mostly flat.
Yes, bring a car seat, just the infant carrier sort that you can strap in with the seat belt, doesn't need to be fancy. It can be checked or possibly used on the plane if you buy an extra seat.
Get a sling - it made the airport and plane and just moving around when somewhere wasn't pushchair friendly so much easier.
Yes to somewhere with a microwave, kettle, fridge - I wouldn't travel anywhere with children with those basics - doesn't need to be a full kitchen. If anything for your comfortable and convenience as you won't want to do every meal in a restaurant with a baby.
Take formula with you (you can pre-order to the airport, but if powder, just buy it and pack it from home, much easier). Take milton's and sterilise in the sink.
The heat will be fine. It's that hot here in the summer the past few years, so will be totally fine. Bring layers (as it will get cooler in the evenings/indoors), stay in the shade, stay inside during the hottest parts of the day. We were in Spain and it was hot, but manageable. We went out most in mornings and late afternoons/evenings, and we stayed in the shade during the heat of the afternoon. Lots of regular feeds to keep baby hydrated.