Don't show the numbers to the test takers, it will only confuse them. But you may want to use them internally.
Balanced keying (using an approximately equal amount of positively and negatively keyed items) is often used in psychometry and is considered good practice. It allows you to approach a topic from different perspectives.
Some sample items (measuring extraversion):
+ keyed: Feel comfortable around people; Make friends easily; Am skilled in handling social situations.
– keyed: Have little to say; Keep in the background; Don't like to draw attention to myself.
With a four-point scale, you'll probably find that option 3 will collect most of the 'neutral' or 'don't know/don't want to tell/don't understand' responses that would otherwise go into the middle option. For analysis, you can just assign scores 1 (strongly disagree) to 4 (strongly agree) for positively keyed items, and reverse them for negatively keyed items (4 for strongly disagree and 1 for strongly agree). If you allow to skip items, it makes sense to use a scale with a mean of 0 so you can count any non-answered questions as 0 while still using simple addition to calculate the total score.
As for the number of categories in your Likert scale: fewer categories allow for faster decisions and faster completion times of the questionnaire, more categories are slower but people will (subjectively) feel that they can express their feelings more precisely.
Something to keep in mind with Likert scales is that they are ordinal scales, not interval scales. It often works quite well though if you assign numerical values to the categories and use them to calculate a total score. There are alternative methods available for working with polytomous items and calculating totals, such as item response theory, taking into account individual item characteristics, but they're not as clear and practical as simple addition.
The main difference between a scale scored from 1 to 4 (or a shifted version: -2.5, -1.5, +.5, +1.5) and your proposed idea of -2 to +2 is the width of the middle interval, between disagree and agree. With a normal (small) middle interval it takes 3 disagrees to offset 1 strong agree (3*2+4=10; mean 2.5); with a larger middle interval it only takes 2.
With Likert scales people (especially introverts) tend to avoid the extreme categories (I did a research project on this, and it was striking how often for the same question people would answer 'yes, absolutely!' in natural language while in the Likert scale questionnaire they didn't choose 'strongly agree). If you give the extreme categories less weight, that could influence the totals.
Here's an example: total scores for a 5-question extraversion scale calculated with both methods and plotted against each other (for 2700 people):
The scores don't fit perfectly and the differences are statistically significant, but for most of the scores it doesn't make a large difference. The maximum difference on the total score here is only 2.5 (and if you adjust for the different ranges only 2). Looking at the score distribution (range adjusted again) you see the same pattern:
So in short: if you want to use a coding scheme -2/-1/1/2 go ahead, it probably won't matter for the end results. The extreme response categories will count a little less, and the range of possible scores differs from the other scheme. Results will differ, but you can expect the difference to be small. (However, I'd go with the simplest scheme).