Re: milling spindle / tailstock size, both look like they are #2 and the lathe spindle is #3. In the specs, the mill spindle and tailstock tapers are given and the third unspecified one is likely the lathe headstock taper.
Anyway--Littlemachineshop should have 2MT milling collets, and you need a drawbar too for mounting the milling collets. The drawbars are usually particular to the machine, so it might have that already, just left sitting in the spindle. If not, you can probably buy a 'real' one, or make a cheap one yourself.
In a milling machine with an Morse-taper spindle, the drill-chuck is for drill bits only
and isn't ever
used with mill bits, fly cutters, boring heads, reamers or any other kind of real mill bit. The reason is that the drill chuck+arbor isn't usually held in with a drawbar, so it will get pulled downwards out of the spindle while you're working. Even if you are only drilling it can also fall out if the hole/arbor is not perfectly clean, or you drill a lot of holes without tapping it back in (upwards) with a brass hammer now and then.
If you want a retained drill chuck for the added security (of never having it fall out while you're working) you can get one
, but the machine probably didn't come with one. The arbors don't cost much and you may even be able to take the drill chuck you have off the tang arbor and put it on the new drawbar arbor, if the drill chuck is removable (some cheap drill chucks+arbors are all made as one piece).
A normal 2MT taper drill chuck arbor looks like this, with a flat tang on the upper end-
LittleMachineShop.com - Arbor, Drill Chuck 2MT Tang to 2JT
A drawbar-type arbor has a threaded hole on the upper end, like this-
LittleMachineShop.com - Arbor, Drill Chuck 2MT Drawbar to 2JT
Both of the above arbors are for a #2MT spindle and both are for mounting a drill chuck (on the #2 Jacobs taper on the short ends) but the way that they are secured in the spindle is different.
The cost of it all is imposing if you consider it all at once, but you don't need everything at once either. Ask here or over on Chaski, say what you're trying to do and somebody will know the name of the accessory that you need to do it. You're just going to have a 'wish list" of little doohickeys for a looooong