To your actual specific questions:
A P/E of 13.6 seems a bargain for a big company like Apple.
Question 1: Would you agree on the above statement?
Question 2: How you pick stocks based on P/E ratios?
P/E gives a quick way of comparing a company to current market/industry valuations/sentiment, rather than a buy signal. I'd be comparing it to competitors - Microsoft, Google, Samsung? If it's significantly different to it's peers then it raises the valuable question of why?
If you're looking for undervalued companies based purely on an earnings valuation then P/E will flag those for consideration.
When referring to EPS in next 5 years, it seems that Apple isn't a good buy. P/E is bigger than growth, meaning it is overvalued.
Question 3: would you agree with the above statement?
Again, raises questions rather than black/white answers, but from this I'd think it likely that the market believes the company has something else going for it besides it's projected earnings growth.
In this case, if we refer to past performances of growth rate, the actual P/E seems to be a great bargain.
Question 4: would you agree with the above statement?
It's certainly an interesting POV. Personally never looked at this valuation. As a general rule though past performance is a dangerous place to look for future outcomes.
Now, to get a decent return based on future dividends, the fair price for Apple should be $30.50. In terms of dividends, Apple is 17.7 times overvalued.
Question 5: would you agree with the above statement?
The return I get from dividends investing in Apple is an absolute "stay away".
Question 6: would you agree with the above statement?
All I'd be taking from these valuations is that you've correctly identified that there's no profit to be had at current dividend levels.
If we take into account the growth rate in earning per share, a fair price for Apple would be $610.0 versus the actual $540.67. In these terms, the actual stock price is slightly underestimated.
Question 7: would you agree with the above statement?
Yes - providing you're RRR is 10% and not 11%, and that you're confident they'll hit the earnings growth projected.
From reading the news I know Apple has experienced exceptional growth over the last few years due to it's largely unique product line but this market advantage has been worn down so growth is not expected to be as stellar in coming years. It has managed to amass a rather large amount of cash though so there's an expectation that the cash will be used to fuel growth through large R&D or acquisition. There's also been a shareholder movement pushing to have a significant portion of the cash returned to shareholders through a special dividend also.
All of this is hinted at by your various valuations and that's what I like about the maths aspect of stock picking. In a matter of minutes you've managed to ask a number of valuable questions that point you in the right direction.
The decision I then have in my mind is after understanding why it's valued where it is what am I willing to speculate on that isn't being allowed for?