Is 'start not to do/begin not to do' grammatically correct and does it make sense?
We would be far more likely to say "I gave up doing/Istopped doing" than "I started to stop doing". You can, I suppose start to cut down on (doing) something, but it is hard to start to stop (doing) something. Stopping is normally an instantaneous thing. The second you stop doing something, that's it.
Context is always important; labelling is rarely important.
Do you mean something like "I started to not smoke ten years ago"? I can understand the concept because, much like alcoholics, many smokers feel that they have never completely given up or stopped smoking, they are just constantly in the process of quitting. On that basis, they may not want to say "I stopped smoking ten years ago" or "I gave up smoking ten years ago". However, I would suggest "I last smoked ten years ago".
Can you give us some example sentences in which you use your construction though, so we can see how you wish to use it?
Real sentences are always easy to work with than fragments.
Let's say you have a few bad habits. You leave dirty dishes in the sink overnight. You interrupt your spouse a lot. You eat when you're stressed. You've resolved to eliminate these things, but you know it won't be an instant process.
I'm starting to stop leaving dishes in the sink overnight.
I'm starting to stop eating for stress relief.
I can see using those sentences.
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.
Oh, I got lost. No surprise.
Yes, I can imagine saying "I'm starting to not leave dishes in the sink every night, but I still slip up every once in a while." or "I'm beginning to not eat for stress relief, but that box of Girl Scout cookies did get me through my father's surgery."
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 learned that the negation of to-infinitives was 'not to do' and 'to not do' was grammatically wrong or informal if it should not be called grammatically wrong.