The one piece of advice I would give if you are planning on staying in Thailand long-term is to get a good job here. That would at least give you annual extensions of stay, and would help if you wanted to go down the route of PR or eventually citizenship.
A 'good' job is one for a company that will give you a reasonable salary and get you one-year extensions of stay and work permit, and not teaching English for a dodgy school for a terrible salary, where you get a visa and "maybe" a work permit, but you have to do a visa run every three months.
The naturalisation process in Thailand is difficult. The requirements are stringent, and require going through many stages and waiting long periods. Also, if you are male and you were somehow able to acquire citizenship before the age of 40, you would be required to do National Service in Thailand.
Also, if you are planning on staying overseas to avoid National Service, I would recommend staying in a country that is itself not very strict about its own National Service.
As for applying for refugee status, it is very difficult to answer as Thailand has a particularly 'difficult' history when it comes to refugees, and have turned away Uighurs (who have horrific things happening to them in this region), Cambodian victims of the Khmer Rouge, people displaced by the Vietnam War, and more recently the Hmongs (many of whom had been born in Thai refugee camps, and were all deported by coach to Laos a few years ago). I would not expect them to be particularly accommodating when it comes to a war that is taking place on the other side of the world. Especially under the current military junta.