DIY Christmas Tree LED Ornament – Adding SMD Switch to power on ATTiny45 and Wiring Tag Connect to Attiny
As mentioned, you could run the whole circuit off the battery. Blue and white LEDs may be a bit difficult on 3V. That being said, if the forward voltage at 20mA is 3.3V, you will probably get some light out of them. 

You could put the mode switch on the reset line, and program the next power up state in flash. A bit hackish, but it probably would work. 

You need to pick the right Tag-connect for your programmer. As Ignacio mentions, you probably would want the 2030 model. I would recommend against using the Tag-connect altogether, as it will easily double your project cost. Just use the existing pin header connector. If you must get the Tag-connect, get the one with latches, as it will make debug sessions easier. 

A gotcha with Eagle is that you can set the net name in the schematic without actually displaying the label. There are probably some connections that are in the file that are invisible because the microcontroller pins aren't labeled. 

Don't forget that you need to match footprints of your new parts to the actual parts you buy. To increase the chances of your board coming out right the first time (and in time for the holidays) I recommend you buy all your parts now so you can make sure they fit on a 1:1 printout of your PCB layout. 

Good luck