Another series resistor might be a good idea if you were going to swap in other motor controllers, but
for this one the input is already current limited and isolated from the rest of the circuit, should be fine.

There are still sources of nixies and driver chips on eBay, mostly russian ones.  I know I found a source of cheapish power supplies
for nixies a while back, 12V in and 150V out.

Sounds like either your alternate power is not stable, or you have shared the ground wire from Arduino to both
supply and sensor - keep sensor ground wiring separate from any ground wiring that takes significant current.

gnd to pls- and dir-, then your steppin to pls+ and directionpin to dir+.  These inputs are an opto coupler with built-in resistor

I think you also need ena- to gnd and ena+ to 5V to enable it.

The schematic for that sparkfun level shifter shows that the RX connections are the ones to use for converting
5V to 3.3V, so try using them.

The switch is a low side, so it's connected to the common ground of a 6v, 1.6MAH rechargeable pack. As I was told, the 60v means absolute max, so says on the fact sheet

OK, plenty of headroom on the voltage spec then

Well I'll tackle a few points:

Current is deemed to flow from positive to negative through a resistor or other "dissipative load" (meaning
something that converts electricity to another form, be it heat or movement or light).

In reality in metals it is electrons that flow (and they are negatively charged so flow in the opposite direction
from what we call current - a historical accident as "positive" and "negative" charge were defined before
discovery of the electron.

In semiconductors charge carriers can be both electrons and holes, holes are positively charged.

Ohm's law applies to most conductive materials from metals to semiconductors (when just a pure slab of
one kind of semiconductor).  The greater the resistance, less current flows (for a given voltage difference).
The greater the voltage, the more current flows (for a given resistance).

In a battery we have chemical reactions converting chemical energy into electrical energy - in effect a voltage
difference.  The battery has its own "internal" resistance, which limits the maximum current should you short
the outputs of the battery - the more current from the battery the greater the voltage loss across the internal
resistance.

PWM has a fixed frequency and a resolution limited by the clock cycles per PWM cycle.

With Sigma-delta the input value is summed to an error register and the output subtracts
from that register - in effect there is a feedback loop trying to keep the error within bounds,
and the input resolution is decoupled from the output resolution.

You can use single-bit output, in which case the proportion of ones in the output (averaged
over time) is proportional to the input value (which can be 16 or more bits if you want).  The
frequency of the output is not constant and usually the signal needs converting to analog
with a very tight-spec charge-summing stage (the precise timing of the clock edges is all-important
in getting good linearity).

But you can also use PWM as the output, say 8-bit PWM out, 16 bits in, and the error value
has its top 8 bits pushed out to the PWM every PWM period - the lower order bits are still
accounted for and provide high resolution at low bandwidth.

In general sigma-delta trades bandwidth for accuracy (clock has to run a lot faster than the
sample rate).

Real converters use more tricks and are more complex, but the basic idea is to maintain
an error value and use negative feedback to keep the accumulated error small.

Put over the center of a steadily turning bar magnet it will read low to high in
a sine wave pattern.

Not a perfect sine wave, not when close to the magnet.  Might limit the precision.  Also magnetic
interference from nearby currents can be an issue without magnetic shielding.

The Austria Microsystems datasheets don't claim that great a linearity BTW - but they use a disc
shaped small magnet.

Not sure what that example is getting at, but so long as the interval is less that 2^32 microseconds
(or less than 2^31 microseconds if you use signed longs) then the simple subtractive test works fine.

I would say first start by reading about DC motors, hobby servos, and stepper motors - these are the 3
commonest types that you would need, so find out what each can do well...

Secondly for all motor types there is the process of sizing (working out how powerful a motor you need),
which means knowing a little mechanics and estimating the torque and angular velocity you require.  In metric
its simple,  power = torque x angular velocity (torque in newton-metres, angular velocity in radians/sec).

Estimating torque without building a prototype can be tricky since there are many sources of friction...

And of course you might want reduction gears to match motor speed to what you want (they reduce
speed but increase torque, and lose a fraction of the power to losses).

It has the clever property that you can often get away with having different voltage devices on
the same bus - so long as the pull-up resistor isn't too strong it won't damage the inputs on the
lower voltage (3.3V) device.  You can also pull-up to 3.3V if the 5V devices recognise 3.3V as
logic HIGH, which an Arduino will.

Another property of having open-drain outputs is that the actual level on the bus is the logic-AND
of all the outputs driving it - only if all are high (which is open-circuit) does the line go HIGH
via the pull-up resistor - can be a useful property for some things.

Yet another property is that if the cable gets crushed and the signal gets shorted to ground
then nothing bad happens to the outputs.

The transistor driving the relay coil is wrong, its wired as "emitter follower", rather than
"common-emitter" configuration (which is always used for switching).  Google/wikipedia those
terms for more info.

If the motor is large you need a snubber circuit across its terminals to avoid arcing
the relay contacts when they switch (unless you stop the motor before reversing
drive).

If you check out some of the newer RS485 chips, you'll see they have the
bias resistors already built-in, but you still need the termination Rs,
http://www.digikey.com/product-detail/en/SN65HVD82D/296-35059-5-ND/3725885

That chip doesn't have failsafe resistors built-in (that would compromise bus fanout), but
has a bias built in to the receiver for inputs less than +/- 200mV

That's a 60V MOSFET, what supply voltage was used?