I'm not sure what you mean by "at about 2010" in the first sentence. The sentence is fine if you replace "at about" with "since about", but not if you replace it with "in about". ("At" can't be used for years. It can be used for prices, but a price doesn't seem to work in this sentence, as it's discussing production levels rather than price levels.)
- I have read somewhere that the production of push-button mobile phones has started to come down since about 2010. -- Correct - although "I read" would also work and would perhaps be preferable.
- *I have read somewhere that the production of push-button mobile phones has started to come down in about 2010. -- Wrong: should say "started" (or "had started"), not "has started".
The following is wrong:
- *Two days ago I have heard one manufacturing firm begin to put out such phones with renewed energy first time in almost 10 years for the reasons of security.
The presence of "two days ago" means that you should use the simple past "I heard". Secondly, "begin" is wrong here, because "heard ... begin" would imply that you had heard them begin (i.e. you were present and had heard the actual production start), rather than that you heard that they had begun. So it should be:
- Two days ago I heard (that) one manufacturing firm had begun to put out such phones with renewed energy for the first time in almost 10 years for reasons of security.