sorry if this is stupid, but when i bought my guitar the strings were fairly loose, and i tuned it with the tuner that came, and the strings were still not very tight(which might be how their suppose to ) but i can then keep tighten them and till the lights say the right reading again but their alot tighter then and higher pitched than before, but both times the tuner was saying the right note name for each string. so i'm just curious of how tight are the strings suppose to be? really tight or kinda tight?
ur supposed to stretch out the strings when you get them and before you put them on. now that they are on just pull on them a lot and then tune it up and keep doing that until it doesnt untune anymore
the way it come (loosened) are the correct pitches for the notes

the tightened higher pitched ones are the same notes but are a octave higher meaning they have the same quality of the note but are higher pitched.
Fender Stratocaster,Yamaha ERG 121 and 5 other random low budget acoustics
good question, the above guy posted the answer to the reason why the tuner recognizes the string as the same note.

There are only so many notes in music, but they go up and down in pitch (also frequency). So you can have for example an A note, and then an A note 8 pitches higher,which is an octave.

As far as how tight your strings should be, they shouldn't be REALLY loose but pretty tight. When I tune my guitar I use my tuner just so that I don't tune them too high, and if you need to, look up "online guitar tuner" in google and use the Gieson guitar tuner just to referance how high or low you are off the real pitch of the note you are trying to get to, despite the pitch your tuner will say.

Hope this helped. If you need to visualize octaves, check out a piano. A piano has all the same notes but goes from a really low pitch to a really high pitch, low being left and high being right.
if the more "loose" of the two choices doesn't rattle all over the fretboard then i'm guessing it is the correct one. it's impossible to be 100% certain without hearing the guitar, but i'm 99.9% certain. i'm quite impressed that you managed to tune all of your strings up an entire octave without breaking any strings.