Sounds to me that when booting from floppy, Startup-Sequence is missing the proper NSDpatch necessary for larger HD's. Very typical behaviour you're experiencing there without the proper patch(es) or OS. Compare the 2 Startup-Sequences (floppy and HD) and see what all is missing.
As far as the computer crashing after 15mins... sounds like a power supply problem or possibly heat related. I'd first try re-seating any chips (ram, kickstart, buster if you have the right tool) or the accelerator card if installed. If you have a barebones system, the 030 is mounted directly to the CPU, so no chance of re-seating that puppy. Also, what kind of Kickstart are you using? Is there a ROM tower installed?
One last thought... for grins, completely remove the HD and see if the computer still crashes after a while. Some HD's take more power than others and also throw out more heat. Things are pretty cramped inside an A3000! Whenever I restore/refurbish old systems, I make sure the power supply and its fan are super clean. That thick dust you'll see across all the guts can cause components to fail prematurely (especially bridge rectifiers and capacitors). Compressed air works well ONLY in conjunction with a vacuum cleaner and toothbrush/paintbrush. Compressed air by itself will just blow the dust everywhere else you don't want it.