I used the dial indicator and pushed the tailstock back the other way the same .183". Maybe that's cheating. I was slacking and should've used a test bar to verify exact alignment.
It may be cheating a bit, but I'd think you'd get it pretty close that way. For your attempt #2 I'd definitely make sure you know that the headstock and tailstock are in alignment when you need them to be. If you have a compound with enough travel, you may be able to set that to turn your taper instead of setting over the tailstock. There's less work getting things back to normal when you're done.
A cut that's 0.040" off the radius is pretty aggressive for a small, light machine - especially when the part is between centers rather than gripped tightly in a chuck or collet. And even more so with the small diameter to length ratio you have with this setup. It's great that you're experimenting and building your confidence, but you've likely bumped into a machine/setup limitation here. Save those aggressive cuts for when the workpiece is very well supported and you're cutting close to the support. There are lots of definitions for this, but one that might work is that you have full engagement the length of the jaws of your 3-jaw chuck, and you're cutting no more than 1.5x the diameter away from the jaws. I'm totally making up a "rule of thumb" there, but that's pretty much my criteria for pushing the 25HP motor on "my" CNC lathe at work to start whining.
Regarding the vibration, see if you can offset the weight of the dog with something mounted to the faceplate. That, or see about a smaller dog that isn't subject to as much centrifugal force. You already figured out part of the trick to 1018 and nice finishes. Once you get things balanced and can get your RPMs back up where they need to be, try taking a bit heavier cut on the finishing pass instead of a few thousandths. You'll probably be quite happy with the results.
There will be deflection that varies depending on your DOC, so to make sure you hit your target diameter, try the following:
Target Diameter: 0.875"
DOC on finish pass: 0.010"
Let's say you start with 1" stock. You rough it with whatever works until you're at 0.915" (Target + two more passes). Adjust to take 0.020 off the diameter (remember, this is the same as 0.010 DOC) and take the cut at the same settings you plan on using for the finish cut. Although you'd think it would measure 0.895 now, it probably wont. It should be close though. Figure out how much you have left and take one more cut with the same settings and adjusted DOC. At this point you should be pretty darn dead on your target diameter, and a nicer finish than you're getting now.
I apologize if you're beyond this point and it sounds a little basic. But as simple as this all sounds, it takes some practice to be able to hit a final dimension without sneaking up on it. I know lots of experienced people who still don't get it. With some materials it works, but with others (like your 1018), the finish goes to hell if you take light cuts.
Unless you're in a garage with decent ventilation, don't bother with cutting oil. It won't help much with the finish, if at all. It will just make a lot of smoke. If you're going to use anything, use water soluble coolant in a little squirt bottle to keep everything well cooled. Don't wait for something to heat up before you squirt it, keep it cool.
Regarding not being able to cut the full length, that's kinda the nature of the beast with between-centers turning. If you didn't have the tailstock quill well-extended, try extending that out enough to get some room but still give good support.
There are so many more things I could go into, but it's probably best if I just let you go figure the rest out for yourself. You'll get it if you stick with it.