When I play the 7th fret (which is E) of the string A, the string E rings. Also the same with the 7th fret of D (which is A) which causes the A string to ring.

Does this happen with all guitars?
try muting the open strings with the other fingers of your fretting hand, or the palm of your picking hand.
They don't call me "The King Of Awesomeness" for nothing!
This is actually guaranteed to happen, at least to a tiny extent, whether you are accidentally striking those other strings or not. Any string will want to "pick up" any nearby sounding resonant note.

This is not to say the other posters are incorrect, though. They are right, it is not your guitar's problem and you should work to perfect your technique: picking the strings "just right" and muting unplayed strings with your fretting or picking hand...or both.