for the shutdown problem, try to check the "disable the fast shutdown":
http://www.putergeek.com/msconfig_98/

Copied from a post I just made for a user that is experiencing a similar problem:

Refer to this link as it contains a patch that might address your problem (Windows 98 Second Edition Shutdown Supplement): https://support.microsoft.com/en-us/kb/239887
I've attached the file (004756us8.exe).

You also might want to install the IDE Hard Drive Cache Package (https://support.microsoft.com/en-us/kb/273017). File also attached.

Alternatively, download and install the Unofficial Windows 98SE Service Pack 2.1a (there is a fan made website that has continued this project by adding other functionality to Windows 98SE and is currently at version 3.44).
It contains a number of Microsoft patches/fixes that should address most of the issues you're likely to experience.
You only need to install the main updates (the rest are optional).

Copied from a post I just made for a user that is experiencing a similar problem: 🤣

I know its annoying but I also know its not deadly. When you stuck with "Windows shutting down...blah..." window, just wait for a couple of minutes and shut down your computer. Nothing bad will happen. I had many rigs that do that to me, and even during the days, when you make one last update or install that one last utility, this was happening, sooner or later. The matter was not if, but when. We used 98 machines like this forever at that times without anything bad happening.

Installing version 2.1 of the Unofficial Service Pack and the Shutdown Utility seemed to have fixed the shutdown and entering DOS issue. I still have the certain DOS games video flickering problem. I am going to see if computer guy nearby has an old video card I can test and buy. Thank you for the help so far fellow users.

I tested another video card and I am still having problems. To my knowledge the flickering only happens with games that use EGA graphics. Does anyone know why this is?

I fixed all the issues I mentioned above. The only thing left is hooking my PC Speaker back. The thing is I do not have the motherboard's manual and no one online seems to either. Is it safe to test all the jumpers until it works including while the machine is powered on? Thanks.

It's hard to hurt anything messing with front front panel connectors. I doubt you could do any damage hooking a speaker up to the wrong place. Got a picture of the header?

Here is a picture of it.

To be on the safe side: With the PC powered off, use a continuity checker to find pin(s) that are connected to 5V. That's one side of the speaker plug. Directly next to it, there should be a pin that carries GND, that's the direction where the plug should reach for the other side.