始める vs 出す vs になる


Elementary grammar books cover the difference between '赤ちゃんは歩き始めた' and
'赤ちゃんは歩き出した'. However, there seem to be one more grammatically correct way to
say 'to begin doing something', namely Vstem+になる. I wonder how that fits into
picture. While '働きになります' probably says 'I will start working', and '働き始めます'
says 'I am starting to work' I am unsure what is the difference between
'働き始めました' and '働きになりました'.

Follow-up question: if I want to say 'He started thinking it is good', I
should use 思うようだ about third person. So これがいいと思うようになりました is fine (and means
'he finally thinks it is good', like maybe I convinced him), and first version
changes to これがいいと思う始めるようだ or 思うよう始めた?

