There are details you did not include.
Is this drive a MBR type?
Linux does not number partitions like Windows for MBR.
Example:
Linux might have these numbers from left to right:
1 2 5 6 3
For that same drive Windows could number therm as:
1 2 4 5 3
Sorry, I do not know how to explain this in 25 words r less.
Rather than do it by hand, you need to use a partition manager that will automatically provide the right information.
Yes, any laptop does have room for a second drive.
You have the USB ports. Both DOS and Linux can boot from USB.
You can use the CD drive for DOS or Linux
DOS 6.22 was made available on a CD sometime ago and it seems that Microsoft has not stopped people from using it for person use only.
Here is howto find the DOS 6.22 thing:
Google put dos 6.22 on a CD download
Does that help any?