Richard:
First of all welcome but please if you have a couple minutes post a thread in the welcome area and tell us a little about yourself. Whats are your interests, and where are you from, and a little about your home shop.
OK back to the question.
Any of those sizes will work with a couple of cautions.
first of all it is important to have the cutting edge of your tool at center height of the lathe. The simplest way I know of is to eyball it close then gently hold a 6" rule between the tool bit and a piece of round stock held in the chuck if the bit is high the top of the rule will kick in if low the bottom will kick in the rule should be vertical. the way height is adjusted is with shims .Place strips of metal under the tool until the correct height is achieved. An easier way to do this is buy a quick change tool holder. These are much easier to adjust for center height they have a thumb nut.
how do you guys cut such small parts on your lathes without bending the rods or metal?
there are several ways of supporting metal first of all think in terms of the diameter of the metal you are cutting. 3 times the diameter stick out unsupported is probably a good rule of thumb in other words a piece of metal 1/2 in diameter should not stick out of the chuck more than 1 1/2 inches without support.
there are three ways of supporting metal being cut
1) hold it with a center in the tail stock you can use a dead center but use the proper high pressure lube and careful adjustment or a live center these have bearings so the part touching the metal spins with the work. you have to drill a center hole in the end of the work with a center bit so the center can support the work.
2) a steady rest: these bolt to the ways and go around the work and support it in the middle or near the end.
3 ) A follow rest is mounted on the compound and moves with or follows the tool. These are used for long thin pieces.
#1 is the most commonly used method
I would like to try my hand at the Mine Engine, but maybe I'm starting off with a complicated project?
If you think the Mine engine is too complicated for your first engine you are probably right.
May I suggest a LMS oscillator
http://littlemachineshop.com/Projects/OscillatingEngine.php
or a Mcabe Runner
http://npmccabe.tripod.com/mccaberunner.htm
Happy machining
Tin
