There are many single sided designs out there, I entered "Arduino single sided" into Google and looked under Images and found these right away:

http://www.theparsley.com/arduino/diy/
http://www.vonkonow.com/wordpress/2012/10/nanino-the-diy-friendly-arduino/
http://electrotormentor.blogspot.com/2011/06/single-sided-arduino-uno-board.html
http://mytronx.com/projects/myduino-v4/

They list three different LS7366 models, each with a different price.

The "R" is a 14 pin DIP, The "R-S" is a SOIC and the "R-TS" is a TSSOP package. It's in the datasheet: http://www.lsicsi.com/pdfs/LS7366.pdf

But there's no reason why a PIC or ATTiny, properly programmed of course, shouldn't work just as well. It might even be better, since the LS7366 must be reset externally (either by a pin or by reloading the counters) while a PIC or ATTiny could automatically reset its counters and start over.

How about here:

http://catalog.2k1.co.uk/acatalog/2001__LSI_CSI_Computer_Systems_Inc_140.html

I don't know anything about the company, it's just the first one I found in a search.

Edit:
Gemini claims to have them:

http://www.geminielec.com/

An LS7366 might fit your needs. It's a 32 bit counter with SPI interface that can be set in non-quadrature mode to count up or down and provide an output on overflow/underflow. I believe one of the pins can be configured as a reset as well.

Also, low frequency crystals require significantly less power to oscilate, which is a handy thing when operating small battery-powered devices such as RTC  chips or wristwatches.

A 74C922 will probably work, if you can find one.

PS/2 keyboards (and mice) have open-collector outputs. You will need pull-up resistors to read them with a scope (or anything else).

You may find this page useful:
http://www.computer-engineering.org/ps2protocol/

Code:

int HighTempA = 0.1;
int LowTempA = 100;

Well there's your problem.

Integers don't have decimals.

You might want to check out the Nixie forum over at
http://www.tubeclockdb.com/forum/index.html

Lots of nixie users over there with far more experience than myself.

Ah, sorry. I read that as an IN-18 tube. Not enough coffee this morning.

Unfortunately, you can't really test a Nixie tube without a high-voltage power supply.

Here's a simple nixie boost supply:
http://www.ledsales.com.au/kits/nixie_supply.pdf

And a driver more suitable for an IN-13 and an Arduino using PWM:
http://www.smbaker.com/experimenting-with-in-13-bargraph-nixie-tubes


Edit: I see you already have a driver and a supply.

For testing, a 5K to 10K resistor from the indicator cathode to ground and a 220K on the Aux cathode should cause at least some of the segments to light when voltage is applied. I would not recommend connecting the cathodes directly to ground.

Your Arduino should be fine if you make sure to keep the anode voltage away from the Arduino and yourself, make sure the grounds are common and output a PWM signal using analogWrite() on an appropriate pin. You may need to adjust the value of the emitter resistor to get the range you want.

A very simple program could be used to test the circuit, something like writing an analog value, delaying 5 seconds, writing another analog value, delaying 5 seconds...

As a safety precaution, leave the the high voltage supply off while programming the Arduino. Then reset the Arduino, turn on the high voltage and see if it works.

I would also recommend you power the Arduino from a battery and not USB.

In the 'old days' we used 7441 chips to control Nixie tubes. As I recall, their outputs were just high-voltage NPN transistors pulling the cathodes to ground. Current through the chip was minimal, limited by the anode resistor. I think the transistors also had zener diodes across them to protect them from over voltage. The more modern equivalant is the 74141.

A quick google search turned up this:

http://web.jfet.org/nixie-1/NixieTransistors.pdf

and this:

http://dangerousprototypes.com/2011/12/09/driving-nixie-tubes/

I want my obituary to read,

"In lieu of flowers or other memorials, his harem has requested that the resulting crater be named in his honor."

From my own personal experience, that's the smartest thing on Newark's site.

True story:

Last week I went to Newark's site to order some parts and logged in with my account, which I've had for several years. I found my parts (not always an easy task on their site), placed my order, confirmed my billing address, shipping address, credit card, shipping method and clicked on "Submit Order."

I waited and waited and waited and waited and...

Their site kicked me back to the start, clearing my cart and all my information in the process and forcing me to start over. All the page said was, "There is an error in your submitted information, please resubmit the flagged items."

Everything was flagged.

After several more tries, all ending in failure, the site finally tossed out a message that I needed to contact Customer Service at their toll-free number to complete my order. Which did me no good as the number simply went to a recording telling me that they were closed and to call back during business hours.

After fuming a bit and discussing my frustration with my wife, I had an idea. I started over, but this time I set up a brand-new account. Different ID, different e-mail address, even a different credit card. (But the exact same name and address.)

This time it worked, although it was rather slow to respond. At the end, I was asked to fill out a survey concerning my experience, which I did. Oh BOY, did I!

Flash-forward to yesterday when I received a call from a Newark representative, who was calling to welcome me to Newark and thank me for placing my FIRST order.

I informed her that it was NOT my first order and then proceeded to detail my recent experience. She seemed sympathetic and tried looking up my original account, which for some reason couldn't be found.

She then asked if I had any other concerns and I mentioned how I often find it difficult to locate parts in their system. She asked whether I was searching by the manufacturer's part number or Newark's part number. I told her, "The manufacturer's, of course."

She suggested I might have more success if I tried searching using Newark's part number.

I thought about it for about half a second and asked, "And just how am I supposed to know YOUR part numbers? I don't think God Almighty knows your part numbers."

There was a long pause and she finally acknowledged, "Yeah, that probably wasn't a very good suggestion."

However, she did tell me that they're rolling out a new website in a few days and it should be much better.

We'll see.

Yeah, I see that now. For some reason I _saw_ the N-channel, but kept thinking of it as a P-channel. Getting old, I guess.

Yes, you will either need to add  a crystal (8 or 16 MHZ, doesn't really matter, just wire as shown on the page you linked)

Or

You can also use an ISP sketch that provides a clock source like this one:
https://github.com/adafruit/ArduinoISP

Which provides an 8 MHz clock signal on the UNO's Digital 9 pin, which then needs to be jumpered to pin 9 of the target 328.