Non-System disk means that the floppy doesn't have the boot sectors hooked up on it. Straight copying a floppy doesn't put the boot sectors on there. Since you can boot a 95 floppy ok, I feel that there is a big possibility that the 6.22 install disks are bagged. You might try nailing a 6.22 boot floppy from here:
http://www.oldstuff.myagora.net/lightspeed/main.html#Boot%20Disksor let me know and I'll email you a self extracting image. I also might have an image'd set of 6.22 install floppies around here somewhere...
BTW - imaging a floppy also cops it's boot partion when you reconstitute.
After you make the disk you can boot that, partition and format, reboot (it will make do that), and your hard drive will be ready.
From there, boot to the new boot floppy, switch to the first 6.22 install disk and type "setup". Everything should chug from there.
The reason I am saying to get a DOS 6.22 boot floppy instead of using the Windows 95 boot floppy is that the DOS versions are different and many times you will get errors "wrong DOS version".
How big of a hard drive are you playing with? Not sure if the install will partition and format for you or not - I always just partitioned up myself. I do know that DOS 6.22 doesn't do FAT32 - so if you try to use a Windows 95 boot floppy to partition and format and get asked about "large disk support" say NO.
Keep in mind that this is an excellent time to get familiar with all those destructive things that the books tell you not to do. Your first "format c:" is very liberating... and playing with partitions for awhile might also be fun. Partition Magic, anyone? Maybe Rannish partition manager?
I had a ball playing with that old IBM 486 I had awhile back - and experimented quite a bit. I encourage you to look back 9 pages or so at MisterWeave Computing for when I was messing with that old IBM - lots of good intel on Windows 3.11, hard drive and bios workarounds, install tips and links...
I hope this helps.