My daughter was quite a good violinist but I could never persuade her that she should actually know the names of the notes she was playing, what key she was in, what the chords were that her melody was based on etc. (I'm a Music teacher btw). It wasn't 'til she took up guitar that these things became obvious. These things are about understanding how music works and make the piece you're playing make sense. A lot of single line players seem to work on the basis of 'a note on that line means put that finger there' - singers often don't even bother with notation at all. I'm convinced that this makes the music harder as you don't really know what you're trying to do.
The best way is to be playing lots of different music with other people, talking about it, reading about it, listening and learning- make understanding part of your music. As I understand it, the two most important aspects of violin tone are good strings and confidence. Stand tall and use lots of bow; make your mistakes right out loud.
Good luck with it.
Right, this is a huge area so I'll try and answer this objectively.
Firstly 'playing by ear' can be really helpful, but only when combined with a knowledge of what is actually going on in the music i.e. the theoretical side of things. Knowing when a note is or is not in tune is half the battle with the violin! If you feel you need to build up you're theoretical knowledge I suggest you take a couple of months piano lessons (enough to know which key is which, at the very least), also I you may find the 'AB guide to music theory' useful. In the meantime go to musictheory.net it is a great site full of basic theory lessons.
There aren't really any easy ways of learning vibrato. I suggest you watch some videos of Vengerov and the like playing a concerto on youtube, paying close attention to their use of vibrato then try to mimic that yourself. Also you can practise rocking back and forth on the tip of your fingers at a table. Note that the key to this is keep you left arm completely relaxed!
Re. positions, have you ever done any sliding exercises? For example placing your first finger on the A on the G string and sliding up to C to find third position? If you haven't, do it. Also you can ask your teacher for go through some etudes designed for changes in position.
Sight reading as you said is a huge topic in itself. I have a few suggestions all of which have helped me:
1. Learn to sight sing - When you know what a minor third sounds like in your head, you know how it will sound on the violin.
2. Know your instrument - Its all well and good knowing where third position is on your instrument, but do you know where every note is within that position?
3. Sight read as much as you can - Play at least one new piece every day, don't learn it, just play it and put it down. This site will be helpful to keep the expense down: imslp.org go to the music library search for a piece, or composer and play.
4. COUNT - If you don't count, the chances are that the music will pass you by before you even realise.
Please note again there is NO quick fix, if you want to improve you have to challenge yourself.
Finally in regard to making a beautiful sound, there is not one singular thing that will make you sound amazing, however when you're playing consider some of these: Are you tense anywhere, especially in your wrists? Are you putting pressure on the bow, remember the weight of the bow itself will produce the best sound. Pay attention to the contact point between the bow and the strings - the best sound should be closer to the finger board on the lower strings and just less than halfway between the fingerboard and bridge on the upper strings. I can't deny that a better quality instrument will produce a better sound, however the same can be said about the bow! A decent bow can do wonders to a fairly bog standard instrument!