My budget is ~$100 and I'm just looking to improve on this stock underperformer!
This is exactly right. You want a sound that pans across the front to not change tonal quality as it pans across the front, and the way to achieve that is by using identical speakers across the front. The next best is "voice matched" speakers across the front.
It is best if they are at the same level (that is, typically, one would want the tweeters at the same level), but some variance is generally regarded as acceptable. I personally use three identical bookshelf speakers for the front, at almost exactly the same height, with the TV above. But one does what one can, and how much deviation from perfection one is willing to accept plays a big part in this.
A better quality center is, all else being equal, better. If it is a poor match for the right and left speakers, then it might be better, overall, to have a worse center that matches the right and left, but otherwise, a better center is always better than a worse center. Of course, there is also the issue of whether something is enough better to justify the price. For that, I suggest you listen and decide for yourself. It is you who will listen, and you who will pay, so it is your opinion that matters on this issue.
Yep, and try to have the same exact speaker for all three across the front. Try to avoid a horizontal center unless you must have one due to space constraints.