Check out bluenile .com. It is a very good website which can tell you exactly how much that would cost. The cost of a diamond depends on the clarity, cut etc. The site will explain.
Princess cut is less expensive than round, oval, pear etc… Assuming it is GIA certified. Just the diamond, spending $5000 not including tax you are looking at .6 to 1.0 depending on the quality/clarity of the diamond.
If she wants a platinum ring that will significantly up the cost. The band itself could be $1000. Platinum is much more expensive than white gold. The difference is it is heavier and is supposed to enhance the diamond.
So if you have $4,000 to spend on a princess cut, you are looking at .6 carat to .9, again depending on the clairty.
There is a link on blunile that will explain what cut, clarity, color means.
you do not want to compormise too much on the clarity and color to get a bigger size, because sometimes you can tell. A smaller diamond which is excellent quality can outshine a bigger one.
In terms of setting, it depends on the person. I would ask her mom, best friend or sister. They would probably know best what she likes becasue setting is very personal. I would ask them to casually ask her in a discreet way, if they do not know already. The setting coud also up the cost depending on what you get. For example, adding the eternity band, side stones or halo will up the cost.
Also keep in mind, I dont think you would be able to get the ring insured if it is not GIA certified.
In terms of ways to propose, I would just say make it personal and memorable. It does not have to be a big “To Do” but do things right and do it in a setting where you both feel comfortable. Get down on 1 knee, ask for her parents permission… if that makes sense in your situation. If she does not like PDA or is private, dont do it on a loudspeaker, crowded restaurant etc.. If she is close to her family, I would consider doing it when they are present. I know it’s the heart that matters, not how you propose, but once you get engaged EVERYONE willl ask you “how did he do it” and you don’t want to say “I emailed her,” “I did it in bed” “I blurted it outi by accident” etc etc (unless that does not embarass you”) because you will have to tell that story many, MANY times…