Re: The spindle bearing play. The horizontal play at 0.0005" sounds OK. The vertical play, at 0.001" may be a little excessive, but I suspect that is not your worst problem and you can live with it for now. On that issue, most journal bearing head stocks are made in two pieces which are bolted together with shims between them. Since the cutting forces on the work piece are generally vertical and in the upward direction, you will usually get more bearing wear in the vertical direction and the upper part of the spindle bearing will wear the most. If your bearing is a two piece with shims, removing about 0.0005" from that shim stack would be an almost perfect repair for that condition.
The saddle is pressed in the downward direction by the cutting forces so it will usually stay on top of the ways. A Vee way will then stabilize it in the horizontal (front-back) direction so movement there is not usually an issue. And it, the saddle, will not normally move upward. Any keepers that are there that would prevent any horizontal or upward motion are of only secondary importance and a rather large amount of clearance is usually OK. The only time that I know of when this is not true is when some kind of milling attachment is mounted on the saddle or cross slide and a milling cutter is mounted in the spindle. The milling forces, when milling in the lathe, can be in any direction and must be carefully considered.
It is hard to tell weather your chucks are OK or in need of maintenance and/or replacement.
Three jaw chucks often have 0.005" of runout or even more. A scroll is not the most precise way of positioning the jaws and wear on that scroll and on the jaws can produce a variety of runout problems. And those runout numbers can vary when chucking different diameter stock because different positions on the scroll are being used. I have an old 3 jaw chuck where the runout exceeds 0.010" in some conditions. I tried grinding the jaws, but there was little or no improvement. I concluded that the scroll and jaws were worn in an irregular manner. I purchased a new chuck. But if your runout is 0.005" or less, a new chuck may not be an improvement.
The four jaw, independent chuck should allow you to "dial in" the centering to any level of accuracy that you wish. But, if the jaws are worn, then a bell mouth condition may exist where they are worn more near the outer edge. This will mean that you can not rely on stock being gripped in the same manner two times in a row. It can tilt one way and then another. Even cutting forces may move it in bell mouth jaws. If it is a brand name chuck, you may be able to purchase new jaws from the manufacturer. If it is a less known or import chuck your only resort may be to replace the whole chuck.
Oh, some jaws are made in two pieces: the lower half which engages the scroll or tightening screws and an upper half that grips the work. In this type of chuck jaw you can buy replacements for the upper half.
Posting some photos with closer views of the spindle bearings and the chucks may help us to make suggestions.