Pretty hard to help with such a small bit of code. Best to post everything, #includes and all.

Are you looking at the right one? On p25, second row, right seems to be the correct pattern, with 0.95 spacing. I would not think 0.8 spacing would be right. Nothing says the lands on the board have to be the same width as the pads on the chip. I don't see why the second one down on the right wouldn't work, depending on soldering method. OTOH, there's no reason not to design your own land pattern that's more suitable to your methods.

I wrote COBOL and GMAP for a few years then switched to all assembler and systems programming.  I wrote some kind of cheesy comm adapter to get SNA and CICS to talk to each other if I recall correctly, it involved a Level 6 but my code ran on the mainframe.  Wrote tons of GMAP for a software company you probably know of since there weren't that many third party systems software vendors for Honeywell.  We did a Tape Management System, a file utility package and a production job scheduler amongst some other things.  I wrote some filsys modifications to add some features to MME GEFSYE that GE left out.  Fun stuff, I really miss GCOS.

I debugged COBOL for the applications guys but never wrote any   Enjoyed GMAP, wrote a lot of that.  My undergrad thesis was basically improvement/mods to filsys saves and restores.  GCOS = God's Chosen Operating System

I was still using a mainframe with actual core memory in the early to mid 1980s.

Yeah, me too in the early 80's; Honeywell.

Cut my teeth on the Series 6000, DPS-8, etc., with the GCOS O/S. Took care of the FEP software for a while, NPS mostly. Did some custom stuff and extensions. Getting dissimilar machines to communicate was quite a challenge back then. Come to think of it, getting similar machines to talk wasn't much easier.

How was the 9V battery connected, exactly? To which pin?

1. Is there any way to test the board to see if it has been damaged? So far I haven't been able to find any instructions on how to do that.

I'd measure current consumption for starters. I haven't measured a Pro Mini myself, but I'd expect 12-15mA with nothing connected to it. If that looked OK, I'd try an ICSP programmer.

2. Is this a known issue that letting a battery run down might create a problem?

Not that I'm aware.

Also just noted the chip seems to get quite hot, which I never noticed before.

That is a bad sign. Is this with nothing else connected to the Pro Mini?

According to the eBay page, the devices are as follows. All very common. I2C addresses are in the datasheets, except for DS18B20, which is not an I2C device.

DS1307 -- http://datasheets.maximintegrated.com/en/ds/DS1307.pdf
AT24C128 -- http://www.atmel.com/Images/doc0670.pdf
DS18B20 -- http://datasheets.maximintegrated.com/en/ds/DS1307.pdf

If you're not already aware, y'all should check out:
http://code.google.com/p/arduino-tiny/
http://hlt.media.mit.edu/?p=1695

There are several others but those are the two I've tried.

There was a post or two on Twitter, so props for that.

In addition to their ADC function, analog pins also function identically to digital pins, so they'd work perfectly fine.
http://arduino.cc/en/Tutorial/AnalogInputPins
http://arduino.cc/en/Tutorial/DigitalPins

As long as pins are available, one button per pin is the simplest and most reliable approach. If a very large number of buttons is needed, then multiplexing or an I/O expander are options.

Just to expand a bit on what Riva said, the DS3231 has a register (0x03) that tracks DoW as a number between 1 and 7. It increments at midnight and rolls back to 1 after 7. It's up to the user to decide which day corresponds to which value and set it correctly.

Personally, I usually use the Time library which does the calculation. Set the time and date, and it calculates the correct DoW. When using the Time library with an RTC, the RTC's DoW register is irrelevant (but not, of course, the date and time registers).

Ok, found PR36MF12NSZF. Will I need more parts to do 220VAC dimmer?

Yes. See the sample circuit in the datasheet for starters. Also, the Arduino will need a way to detect zero crossing, then delay turning on the SSR until some later point in time to control brightness.

Good luck. GIYF. Keep one hand in a pocket so you don't electrocute yourself.

Sorry. So any other SSD relay I can do this job with?

Yes, a random-phase, i.e., non-zero crossing type. At the top of the datasheet it lists some possibilities:

∗Non-zero cross type is also available. (PR26MF1xNSZ Series/PR36MF1xNSZ Series)

So is there a way I can make an AC Dimmer with PR36MF2xNSZ and arduino? What parts should I use?

Umm. See Reply #1.

A triac, which switches the load, has the characteristic that once it is triggered into conduction, it continues to conduct until the load current falls to some near-zero value.

The way a triac is used in a dimmer is to switch it on at some variable point on the sine wave. When the current drops to zero (when the sine crosses zero), the triac turns off, then the triggering circuit repeats the same thing on each half-cycle. Where on the sine wave the triac is turned on determines the amount of power delivered to the load and hence the brightness of the lamp.

This process generates a lot of EMI due to the sharp step in current when the triac begins conduction. Usually lamp dimmers have filter circuitry to reduce the EMI generated.

Another way to reduce EMI is to turn the triac on immediately after the zero-crossing point on the sine wave. However, this means that the triac will conduct for the entire half-cycle. It also means that this type of circuit can only be used for on/off control.