I prefer taps in a normal drill chuck, if they grab they will turn and not break.
For the headstock I bought a collet chuck from J & L recently, it will mainly be used in the CNC but can be put in the manual as well, just grip it in the standard chuck.
All of my collets are ER16 or 32 so that is why I got the 32 for the lathe. Have loads of DA collets as well but all the holders are Cat/BT/Int 40 which fit the Beaver Mill so never use them
For spindle you want a backplate mount or you wont get anything down the spindle, I use mt3 for milling. I didnt want to pay 60 quid for a backplate so I got another cheap mt3, cut the taper off, and shrink fit it into a 100mm lump of steel, and recut the taper.
Tailstock I have a couple of very cheap er11 mt2 with centre and drills always fitted - saves messing with drill chuck when Ill be using a boring bar after. Will be making a floating tap /die holders when I get round to it.