Salam Brother,
Glad to see you are making a trip and May you succeed in making the best out of it.
1: July start is good time to go.
2: Monsoon will not be a problem however you will probably need to bring it to account as an emergency measure.
3: Stay depends on your route, Jaglot is a good place, Gilgit and then towards Hunza you will have plenty of reasonable options to go with.
4: 20,000 per head should be reasonable.
i will be travelling towards Skardu during the same dates so whatever information you may require, i will be happy to share with you.
Regards,
Aaxer.