I have seen similar problems with both RAM modules and CPU clock speed problems.
Start by trying different RAM modules. I have seen a number of motherboards that are VERY finicky as to what RAM will work on them.
Also, on the subject of RAM, for W98 you do not want to have more than 511MB of RAM. There is a bug in the core code that can be triggered with 512MB or more RAM and cause all manner of problems.
Verify that the front side buss and clock multiplier settings are correct for both your RAM and CPU.
I have seen a number of motherboards that under-clock the CPU. This can also cause some strange problems.
1 final thought, is there any hardware that you salvaged from the old system?
Modems
NIC
Power supplies
Even drives can be damaged when a power surge takes out the motherboard.
Remove as much old hardware as you can, and try again.
For that matter, I have found that it works best for W98 to install the OS with minimum hardware.
Then, shut down and plug-in your additional devices 1 at a time.
Chas