Designing a custom arduino pro board
Are there any tiny parts that I can add to this tiny board that could be useful?


That's difficult to answer without knowing what you plan to use it for. Without knowing this, here's what I'd add:


An external crystal, either 16MHz or 20MHz. The internal RC oscillator is not particularly accurate and needs calibration. Also, with the ATmega being a rather slow chip, I would want all the cycles you can get your hands on.
A regulator and adequate decoupling. These are not optional. A 5.1V Zener is not a regulator and your single unmarked decoupling cap is insufficient. Use a small LDO, and read the datasheet to determine what capacitors it requires. Then, each chip should have its own decoupling cap, typ. 100nF, and probably a fair bit more for the shift register.
A header for the UART would be nice, in case you want to connect via serial at some point.
I would replace the shift register with an I2C LED driver (saves some IO pins), but that's personal preference.



  And last but not least, are there any faults in this circuit diagram?


Yes. There are both electrical issues and diagrammatical issues:


Most of your parts are missing values (C1, R1-R11, T1, R10, R11, ...)
You are using a net label for ground in two places. Use the ground symbol.
You have net junctions where you don't need them, and no net junctions where you do need them. I.e. you don't need net junctions when a pin and net meet, but you do need them at "T" junctions. IIRC, Eagle adds them automatically, so this suggests you were not using the net tool, instead using the line tool or something.
A 1k resistor and a 5V Zener is not a good power regulation scheme. Use a real regulator, and make sure to decouple it properly.
Use a header symbol for the ICSP connector.
Your buttons need pullup/pulldowns, and if it were me, I would not route them to analog pins, instead reserving the analog pins for other use.
Why are ADC6 and ADC7 tied to VCC/GND, respectively?
The refdes T is typically reserved for transformers, use Q for transistors.
PB3-PB5 need not be reserved for ICSP, you can break them out as well.
All headers are missing reference designators. These will be important if you plan to make this into a board.