WalikumusSalam..
Great to know you're planning a trip which means we will get an awesome thread later I will try to answer your questions:
1. How long it takes to reach Astore from Chilas?
It takes approximately 4 hours.
2. How long it takes to reach Tirshing from Astore?
About 3 hours (Its a jeep only road)
3. How long it takes to reach Sheosar lake from Astore?
Astore to Sheosar Lake would take about 4 hours (Jeep only road).
4. How long it takes to trek from Tirshing to Rupal base camp?
Don't know about Base camp but it took about 1.5 hours to trek from Tareshing to Rupal village. I'm sure base camp would be much more ahead.
5. What should be the best route starting from Chilas to avoid unnecessary drive. It should be Rama then Astore then Tirshing and then to Chilam Chowki?
The best route would be.. Chilas --> Thalechi (Bunji More) --> Astore --> Rama, Rama Lake --> Back to Astore --> Gorikot --> Two river junction (if I remember correctly, they were chilum nala & rattu nala) --> Tareshing --> Back till two river junction --> Chilum Road --> Chilum chowki --> Sheosar Lake
6. How is the road condition to Rama and Tirshing?
Astore to Tareshing is pretty much Jeepable only.
7. I want to camp at Rama, Trishing and Sheosar lake? Is it possible?
Possible. Last year we camped at Rama, Tareshing & Bara Pani. At Tareshing, you can only camp in hotel premises by paying them a little.
8. Do you have contact information of someone at Tirshing? I need understanding of logistics for trek to Rupal base camp
As far as I remember, mobile phones were not working in Tareshing. I will ask my last year travel buddies if anyone got lead in this regard.