Hi,
I am rather confused with the tense of become. If the evenrt happened in the past, it's clear to me that past tense shall be used. Such as: Tom became a teacher last year. Yes.
But how about following: I'm not sure about the nature of your confusion. I'll just comment on your examples.
The Internet becomes a giant public library today. This tells us that this is planned / scheduled to happen later today.
The Internet became a giant public library today. This tells us that this today, earlier than the time of speaking.
Are you sure you mean 'today', or do you mean something like 'These days'., which means 'in current times'?
You also should consider Present Continous', eg 'The internet is becoming like a giant public library (these days)'.
She becomes old. It's hard to think of a normal kind of context where you would say this sentence.
Again, I wonder if you mean 'She is becoming old'.
She became old. This tells us that this happened in the past.
The weather becomes warm. You could typically say something like 'The weather becomes warm every July'.
Again, 'The weather is becoming warm'.
The weather became warm. This tells us about something that happened at some time in the past.
It might help you to consider a simpler example.
Do you know the difference between 'She cooks dinner' and 'She is cooking dinner'?
Best wishes, Clive