I visited Tolipeer and Banjosa on 16th May 2015, so here what you need to know.
1. First of all, is it ok weather wise or should I delay it.
If anything, the ideal weather for Banjosa is spring...even late spring. Banjosa is a lot lower than Murree/Ayubia and on a hot june day, Banjosa will require AC/fan though still far cooler than Islamabad or Lahore. Whether you want to go there now or in July is up to you. Just make sure your travelling window doesn't have rain as the route could be dangerous when wet.
2. Is there any hotel/rest house at Khai Gala / Toli Pir?
Khai Galla or Chota Galla are not picnic spots so staying there would be boring. Your best bet is to reserve the forest/pwd or army rest house at Banjosa as they are the best. Private ones at Banjosa are not so good. There is no hotel at toli peer but you can find rest houses a few km before toli pir. It could also be a place to stay but I think Banjosa is a better option.
3. How about road condition? Which route to be followed? How much traveling time?
Road condition is mixed. I disagree with the above poster that they are "excellent". They are NOT. Use the Kahuta road from islamabad highway to reach kahuta and take a left for Rawalakot (straight will take you to Kotli). Road till Kahuta is under construction and will require you go quite slow. After Kahuta the road is metalled till rawalakot. THe initial portion is very good and passes through good alpine forest but later you'll encounter several bad patches, particularly as you near Azad Pattan and join the river Jehlum. Road on AJK side (i.e. Azad Pattan till Rawalakot) is slightly better except the first few kms but the constant curvy road with some bad patches means you drive around 40-50 km/hr, which is why it takes so much time. Otherwise, rawalpindi to Banjosa is a mere 110km. From rawalakot till khai galla is also under construction (unless they sped up and have carpeted it by now which is unlikely) and requires slow speed. Khai galla is the point where you turn right to go to banjosa OR left to go to toli peer. Khai galla till banjosa is so-so...a carpeted road but in bad shape. Khai galla till toli pir is a narrow, almost single track road that you drive at around 20-25km/hr.
Some advise: The roads are bad but that only means you need to go slow on them not that they require 4x4. Definitely your cultus (if in good condition) and driven by an experienced driver can go to both Banjosa as well as toli pir. I went there on a Corolla with ease. You'll definitely need A/C till rawalakot but may switch it off afterwards both because whether gets a bit better but also due to the steep climb. Skip tatta pani as it is not so charming place. Avoid travelling on Banjosa road and toli pir road during rain as it can get slippery due to mud and steep incline. While driving on hills, control your car by employing low gears rather than brakes (while descending) or they become hot and unresponsive.
Have fun.