To answer your questions....
#1 Can the Amiga 3000 run without the daughter board?
For OS2.x and OS3.x the Zorro riser board must be fitted.
#2 without a floppy drive?
Yes, though I think you end up with a DF0:

icon on WB.
#3 without a hard-drive?
Yes, you'll eventually end up with select kickstart screen (V1.4 ROMs) or the insert disk screen (OS2.x or 3.x).
#4 without a mouse?
Yes.
#5 without a keyboard?
Yes.
#6 without the battery?
Yes, the RTC doen't need to be fitted either.
#7 Can you buy the weird 15 pin molex type power connector anywhere to make a longer power supply cable?
Haven't seen them for sale, but your best bet would be Digi-Key for small volumes.
#8 Is there a way to test for bad roms?
Put them in a known working computer. It's incredibly rare to have bad ROMs, unless someone has dome something stupid with them (bent pins, wrong polarity). Swapping positions of the two ROMs won't damage anything, but the computer won't boot as every 32-bit word read from the ROM pair will be swapped.
#9 Can you use other Amiga Model parts to use in the 3000, CIA's, Denise, Paula, etc....
Yes, a majority of the parts are swappable with the A500 and A2000.
#10 Can you use other Amiga Hardware, say an A500 internal floppy on a A3000? (I don't care if it mounts or fits (we know it wont) I just care if it would actually boot a floppy)
Yes, the floppy interface on all models is the same.
Also...
The computer doesn't require any fast RAM to boot.
Reseat all the socketed ICs, socket connection problems can cause non-booting and unstable operation.
Check you have correct clock distribution throughout the computer, this mainly comes from U103.
Check the various reset lines are high (5V).
Check FAIL_ is high (5V).
Try disabling the FPU using the.
Check the ROMs are being read by using an oscilloscope, measure on the ROM output enable pin, 20, going low during most of the CPU read cycles.
Check you have a valid TICK signal at J350.
There's various other things you can check too, but I'm running late for work. Look on my webserver for more documentation and schematics.
http://amiga.serveftp.net