1. I want to grow a cat.
2. I want to raise hair.
3. I want to raise a cat.
4. I want to grow hair.
Is it possible to say like 3 and 4?
I don't know about #4. You don't grow hair like corn or rice. I want to grow MY hair OUT. I want to let my hair grow long.
I'm not a teacher, but I write for a living. Please don't ask me about 2nd conditionals, but I'm a safe bet for what reads well in (American) English.
I don't think you "raise" cats, unless you are ranching them for food or fur.
Please note that I'm not a teacher.
"I've stopped shaving my head and started growing my hair." For me, the important part of the second part of the sentence is the word "my".
I want to start growing hair.
I want to start growing my hair.
Remember - correct capitalisation, punctuation and spacing make posts much easier to read.