When you guys play notes with your left hand, do you barely press down on the string? does this increase speed and accuracy?
press down hard enough so that the string rings cleanly without any buzzing or accidental harmonic overtones. if you're worried about your speed, you should be more concerned about how much practice you have than how you fret something.
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If you press a bit harder down on the string you sometimes incurr a vibrato effect. This can be useful sometimes when you are playing faster and you do not have time to oscillate the string much.

Having a higher action is a factor of whether you need to press down harder, with lower action (when the strings are very close to the board) you need not press down hard at all.

As long as you are playing the piece accurately and loud enough then you need not worry about doing so too much.

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You only need to press the string down hard enough that it makes contact with the fret in front of it. Buzzes happen when you have an incomplete connection where the string essentially rattles against a fret (usually the one closest the one you're fingering). If your guitar is set up right and you have half-way decent hand position, it shouldn't hurt, so how hard you push shouldn't really be an issue.

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