With regards to hair, tint is sometimes used to be a more discrete way of saying dyed, or it can mean that there is just a faint amount of colour being added or just a few strands coloured. Also, older ladies may be more inclined to talk of 'tints' but it sounds pretty old fashioned now. The vast majority of people talk about dying their hair.
1.1.- I’m being tinted. (a) Does the verb tint include itself the idea of hair? (b) Can I say: I’m being dying and the listener understands that I’m talking about tinting my hair?. b would have to be I'm being dyed, wouldn't it. However both sentences sound strange - we tint or dye hair not a whole person. So 'I'm having my hair tinted/dyed'. It's possible though, if you were already in the middle of a conversation about what you were having done to your hair, you could say 'I'm being dyed red later' and the person would understand.
1.2- I’m tinting. (a) Does the listener understand that I could be included in the action of tinting? That is, I can be tinting my neighbor or me, i.e., but I can be included in this possibility. (b) The same question as in 1.1 (c) If I want to clarify the point that I’m included, can I say: I’m tinting myself. You would say I'm dying my neighbours hair or I'm dying my hair.
1.3 (a) What a woman (or men nowadays) would say preferably in a phone conversation with a friend: I’m being tinted or My hair’s being tinted or I’m having my hair tinted. (b) Can that woman use “to dye”? They'd say 'I'm having my hair dyed'.
1.4.- If I want to explicit the color, can I say: I’m being tinted of black or My hair’s being tinted of black or I’m having my hair tinted of black or I’m tinting myself of black. Are all these correct? My hair is being tinted/dyed black. I'm tinting/dying my hair black. No 'of' required.
1.5.- I’m tinted every month. (Is this sentence correct?) I get my hair tinted/dyed every month.
1.6.- My shirt’s being dyed. (Not tinted?) Yes, cloth is dyed.
1.7 Is “She had put red tints in her hair” the same as “She tinted of red". To put red tints in her hair would mean that she had red highlights put in ( a few red strands here and there).