#1
On my acoustic guitar when I hold down on the strings the harder I push the higher the pitch goes, is this normal?
#2
That is exactly normal.

What you're doing is pushing it down so hard that it tightens it from the bridge, increasing the pitch.
My God, it's full of stars!
#4
This condition is worsened if you have a rather high action, and if you have large "jumbo" frets. Some players use this as an effect....effectively "bending" the string a bit. Some have experimented with scalloped fretboards for this purpose.
You have to develop a touch that's sufficient to produce a good tone...And no more.