Congrats on booking the first international trip!! I'm no expert on Emirates or kids but here are a few things for you to consider:
To start with I'm not so sure that joining Skywards makes any difference to selecting seats, I'd nearly be 100% sure that you can enter the PNR into the EK website and select the seats that you want within the cabin class. Although I also would have thought that the travel agent would have access to seat selection through the GDS as a colleague of mine recently flew EK and the travel agent selected the seats for her no worries. If you get your PNR (Booking Reference in other words) and enter your details at:
https://fly.emirates.com/MYB/MMBLogin.aspx you should be able to select your seats without being a skywards member.
Something else to consider is instead of joining skywards and if you depart prior to the 31st of March you could claim any flights within March for Velocity points instead (the DJ/EK freq flyer deal finishes on 31/3/11).
Next thing is the seats. On the EK 773's they are configured 3-4-3 but at the rear they are configured 2-4-2 in the last few rows. Given that there is 5 of you and that you have 2 little ones this is my suggestion, and it is only a suggestion others may have a better idea:
Reserve the last row of 3 and the first row of 2 on same side of the aircraft, these appear on seatguru to be rows 45 and 46 for all EK 773's. This way you are all together, you only have to step over family to get to the aisle, you get 2 window seats for the kids, or Dad and Dad in law can sit together for a chat while you and the kids stretch across 3 for a snooze. Downside of sitting up the back is you may miss out on the first choice off the menu but the kids meals will be served first anyway so that wont bother the little ones.
Being about 3 rows from the back the toilets won't really annoy you, the galley may be a little noisy but not too bad, and when you do need to use the toilet it's close enough to not have to bang your way through the narrow aisles that EK have.
Some other things to think about: for an adult in economy on EK the pillow and blankets are fine. You won't find them at a 5 star hotel but they will do the job. For the little ones I'd consider going to the local Kmart buying the kids a couple of cheap pillows and some Thomas the Tank Engine or Barbie pillowcases so the kids can curl up on them. When you get to Vienna remove the pillowcases for later and leave the pillows in the nearest bin. When about ready to leave Europe find a shop like Kmart, buy a couple of cheap pillows again and re use the pillowcases. This way saves carrying the pillows around and makes the kids that much happier when they have something they enjoy like Thomas, and a big comfy pillow to settle down on.
Also don't forget to pack in the carry on luggage the kids PJ's. Whilst adults can sit for hours in clothes, I've seen plenty of times kids go nuts on long haul flights only to settle down to sleep once they are in their comfy PJ's.
Hope all this helps and have an amazing holiday!
Boof