Need help finding PCB components on DigiKey
General rule: most resistors aren't critical, 5% or 1% will be fine (1% resistors generally cost the same as 5%. Why buy 5% parts?). Composition isn't too big a deal either, most are thick-film, and that will be fine here. Power rating is also not important - 1/8W or 1/4W will both work, use what is easier.

If something isn't called out as requiring a tight-tolerance part, it's probably just fairly flexible (this is assuming competent design, something that can be somewhat hard to find in the arduino communities at times).

Capacitors should generally be a decent dielectric - X7R/X5R. The cost over Y5V is negligable, and you get better thermal stability. NP0 is overkill.
Generally, for Voltage, you want to allow some headroom. I like to overspec by 50%, so since the maximum voltage here is 5.5V (for the ATmega), 10V caps would work fine. 16V is probably more easily obtainable.

D2 and D3 are both fairly low-spec generic diodes. D2 can be tiny, frankly I'm not even sure why it's there in the first place. D3 just needs to be able to dissipate how ever much power your device is expecting to draw (it's reverse-bias protection on the power input). 

Voltage rating is just something sane. I'd go for 20V minimum parts, just because anything lower is fairly exotic. Really, just look for what is cheapest that has > 20V ratings and can handle the current you need. 

SMT LEDs are a pain in the ass - there are LOTS of packages, generally each manufacturer has their own. Critical here is the fact that it's a side-firing LED. Ideally, get measurements from the board file, and plug those into the digikey search.

I'd bet "J1" is the "+" and "GND" pads on the lower right-hand corner.