Lets go by your priorities:
- adventure trekking and hiking
Since u r travelling solo so best option is Galiyaat region. Crowd is found in main towns and the moment u leave the towns and hotel/commercial areas, it will be all peace and solitude. Even a 10 min walk away from the road does the trick
There are quite a few proper treks in Galiyaat region and please keep a guide with u so that u can trek safely and don't lose the way.
Try out these treks:
- Miranjani peak
- Mukshpuri peak
- Pipeline Trek from Dunga Gali to Ayubia and then onwards to Khaira Gali
- Lower Topa to Patriata Trek
- Jheka Gali to Bhurban
November is much calmer than summer in Galiyaat and these treks are even more calm and peaceful. You won't find peak greenery anywhere these days but all these treks are through dense pine forests and grassy areas unlike upper North of Pakistan.
Lower Kashmir also has some short treks like
- area ahead of Pir Chinasi
- Ganga Choti
Neelum Valley is only good for travelling as the treks there are very hectic and involve multiple days. If u want to travel and explore local areas then u can visit Neelum otherwise your best bet with mild trekking and hiking is Galiyaat.
1- Places to go for an adventure like Trekking and Hiking (I heard a lot about Nathia Gali) ?
Answered above. There are beautiful villages below Nathia Gali and u can explore those as well where hardly anyone goes.
2- Is it safe to go solo for the very first time ?
No safety issues in Galiyaat
4- Nice and affordable Hotels for stay in between traveling.
It is not peak season so u can easily find hotels on reasonable prices except weekends so try to keep your booking from weekdays onwards.
5- For now approximately the budget is 50 to 60 thousands (Suggestion needed, I am traveling alone with this budget ?).
Your budget is more than enough for 7-10 days trip.