Can You make other photos of this motherboard? From what I've found it has integrated graphic card Ati Rage 3D. Maybe You need to change settings in Bios to be able to use external Graphic Card...

Sorry should've listed my troubleshooting steps.

Disconnected everything.
Plugged VGA directly into motherboard.
Put one stick of memory into DIMM 1.
Replaced CMOS battery.
Plugged in keyboard.
And still nothing.

Here is a photo of the motherboard: https://photos.app.goo.gl/8s2H1pgX72gXYMHy2

EDIT: Also tried 4 different sticks of memory.

Have u tried RAM in dimm2 slot? Is your psu in good condition? Do u have another cpu to try? Can u get into bios?

Well there are many things that can be the problem.

I think there should be jumper for the integrated ATi video, that can be turned on or off.
Also are you sure the parts are you using with the board are tested and working same goes for the mobo.
Also disconnect all parts like HDD, optical drives and such, and try to run only the memory, CPU and Video.
Can you enter in the bios or not.

It shows HP, so this is ASUS OEM board, and OEM parts most of the time are so annoying.

I've tried disconnecting everything. Tried running memory in slot 1 and 2. Also tried another CPU. I can't get into the BIOS. There's no option to do so, and I can't find a clear CMOS jumper anywhere on this board.

You don`t need jumper, just remove the damn battery, for me that does 95% of the time the job done..
Are you sure all parts that you are using is working or ?
Slot 1 is one of my favorite platforms, one of the bad sides on that platform is the damn SDram and some times is hard to find working memory for the board im using, so will be good for you to find more memory sticks to try. Also look the memory sticks contact pins you can clean them with 90% alcohol, bcuz i have seen so much crap, some times that do the trick.

Most interesting things on your case is the mobo if booting, but seems stuck, most of the time they don`t work at all....

Most slot 1 MBs are using jumpers to set the proper CPU speed, i don`t think ASUS boards did have automatic CPU detect.

Is your board stuck on that screen i mean, can you get some beep sounds, or to peress tab and see the real post screen, not only that damn HP logo ?
Maybe if you can`t the bios of the mobo can be corrupted or something.

If you have other slot 1 processors put something that is on different speed, for example 200 or 300MHz

I changed out the battery. I tried a PII 266 in there as well. I just purchased all of these parts so hard to know what is working and what is not. I will try cleaning all of the RAM and see if that helps. Of course this board doesn't have built in sound and I don't have a sound card yet so I haven't heard anything while booting.

Ok.
From what I see on attached photo, near battery is drawn table with CPU Bus Ratio. Check these settings with connector below battery (I think this is for setting CPU Bus Ratio).
On right side of battery are switches for disabling/enabling internal GFX card. Above battery is table with description for doing this...
Try to find "Intruder" Connector. Many OEM motherboard have this and sometimes You need to add/remove jumper on them. But I think in this case mobo shouldn't power up...
In right bottom part of mobo You have PC Speaker so You should hear some beeps from mobo.
What chipset has this mobo? On photo it's covered by CPU heatsink... The best option is to add SDRAM with the same FSB as CPU.

Well disconnect the power supply form the power remove the power cord, also remove the battery for like 5-10 minutes, just like that to erase all possible previous settings.
You can get speaker from PC case, and connect it to the mobo. Well in order to know what is working you need to have secondary parts and to test every single new component you have get last days on them, that is how you will know what is good and what is not. About the HP logo, there are some ways to enter the bios many pressing of the keyboard delete key, or F2, or F12 depend you can try them all. Also you can press many times the keyboard tab key to see if will show the post screen.

FSB is set to 100 MHz. All sticks I have are PC-100. I managed to get it to beep 4 times. Not sure what failure that was, but rebooted and it no longer beeped that error. I've tried hitting F2, F12, and Delete to no avail. Here are the pics of the chipsets:

https://photos.app.goo.gl/2BQTOWI90JIUylxG2
https://photos.app.goo.gl/Ftpy3fh8tQTpVPY03

Chipset which You have there is 440ZX. FSB 66 or 100MHz.
In Ami Bios 4 beeps are for System Timer Failure.
If it beeps 1 long 3 short then:
- Ami Bios gives "Conventional/Extended Memory Failure"
- Award Bios gives "Keyboard Controller error"
It's hard to say what error You may have without any PCI POST card...

Yea it only gave me that tone once so I'd chalk it up to memory not seated correctly. I'm thinking this motherboard is toast.

How are you sure it's not bad ram? Is there anything behind the splash screen (clear it with tab, esc, space, enter, etc)

The AS2830 above the DIMM slots--what's the voltage read probing the center pin? (GND could be left pin, mobo standoff, atx black, fan black) Not sure if that could be for the integrated GPU or the ram.

I think I may have found the culprit. This cap was hiding under the processor and appears to have leaked. It's hard to see in the photo, but you can kind of make it out near the fan header.

https://photos.app.goo.gl/rCzM1Z6lhEvHymrm1

It could be only a fan noise filter. Won't hurt though to take it out measure it or replace it

Have to figure out what cap it is and give it a shot. Don't have a multimeter at the moment to measure unfortunately.

You may also want to try cleaning the CPU connector AND the CPU slot connector on the motherboard.

Having oxidation on either of those can cause problems up to the point of not even POSTing.

For the CPU, a good cleaning with a pencil eraser will work fine.

For the CPU slot on the motherboard, the best thing I have found to work and have used for years on similar slots, including consoles, is a folded up piece of printer paper.
Fold it enough times so it has some resistance when sliding into the slot.

Most likely, once you slide it in/out a few times you will end up with black lines on the paper from the paper taking some of the oxidation off the pins in the slot.

Keep doing the same thing until clean paper comes out with little to no gray lines.