Deosai.may be closed in MID sep...even if it is not closed you may not be able to enjoy the full beauty of the place as the greenery will be all gone or faded. Muharram days are not a suitable time as there are a lot of majalis and processions going on in gilgit baltistan
And regarding your questions
1. Astore road is quiet bumpy , cars can reach but you have to drive very carefully. As far as Astore deosai road is concerned , people do take their cars there but it is not advised at all
2. the easier route is Jaglot-sassi-shangus route but due to road construction work it takes a lot of time to cover that. Astore Deosai route is more beautiful with more places to see in between such as sheosar lake, deosai, bara pani , shatung nala and satpara lake
3. Is this budget for the whole trip i mean hotel, fuel, meals etc??? You may have to add some more
4. Yes you can park car in Astore and hire 4x4 on daily rental basis
5. Answered in the starting para