IMO a cheap pump will do fine. As long as you can connect it to the existing lines, the metering and flow of the oil as all handled downstream (provided nothing is plugged or broken). Hopefully the mill didn't see a lot of use without it, but it could just be that someone pulled it to fix another mill.
With the collet key, our mills always had them and I was taught that they are not a positive-stop to keep the collet from spinning when cutting, but to hold it in place while you tighten the collet, since there's really nothing to hold onto past the nose of the spindle. Once the taper makes contact, that's what is going to lock the collet in place. They are also there to locate the collet so they always load and clock in the same orientation, which might matter if the spindle or collet has wear and run-out.
The bolt with the wheel on top of the ram looks like something the last shop put on for something, but there's nothing functional to the mill about it. Tool racks, DRO arms, an lifting eyes get mounted up there often.
6 thousands back-lash is really good IMO. You might get better with new screws and nuts, or near perfect with ball screws, but it isn't nessisary. Always find your position from the same position and they will be accurate. IOW, if you find your postion on the left, then move to drill a hole on the right, then need to drill another hole somewhere back on the left, go past it a little and indicate moving from the left again. A lot of using a mill comes down to how you lay out and find your positions. Often you will find a corner and make that your datum for everything. When I'm using a manual knee mill, often the first thing I will do is sit down with the drawing and determin all my positions from one corner, not center of one hole to edge of another hole, etc.
Have fun with it! Even a worn out BP can make decent parts. Just keep it clean and oiled and most of the rest is all about having rigid set-ups and the right speeds and feeds.