It sounds like the buzzing on one string would be the action, which is the height of the strings off the neck. You said it was a strat so I'm thinking it probably has 6 saddles. The string needs to be raised off the neck a little bit more. You could try that, you'll probably need an allen wrench to do it. If that doesn't get rid of the buzz on that string, then take it to a pro.

As far as the intonation goes, most of the time you can hear it by ear. Play a harmonic at the 12th fret on each string, and play the note at the 12th fret. These notes should have the same pitch, you will be able to tell if it's really sharp, or flat. If it's really out of whack, take it to a pro. However when you get it back, check it the same way, and you can hear how it sounds for future reference.

Let us know how it goes, and if we can help you with anything else.

I play with an amp quite a bit myself, don't always want to hook everything up when I pick. I'll tell you a little trick that I found out about by accident, if you put the headstock on a wall, or a wood surface (table, desk, side of a bookshelf) it will give you more volume. You're passing along the vibration from the guitar, so it increases the volume a little bit.

Play the strings just holding the guitar, and then put the headstock on one of those surfaces, and listen to the difference. I don't know what this would do if you were hooked up to an amp, but it can help you hear more when playing without one.

JD