What's causing my laptop freezing on boot-up (Diagnostic Help)?

OK computer experts, time to give you a little challenge here! I have a laptop that was dumped on me from my brother in an attempt to fix it. Unfortunately, it's something fairly major that I can't figure out & I need a little help figuring out what exacting is going wrong with the system (which means it's something major). Here is what's going on...

When I turn on the laptop in a "cold boot" (shut down for at least 24 hours, all components at room temperature... around 60 - 75 F), it occasionally freezes up during the start-up process. This leaves the laptop's processor exaust fan running, but it leaves the screen completely off (LED backlights are off). Sometimes, the computer actually gets further in the boot-up process, so I can access the BIOS or even as far as to log into Windows. However, at some point, the entire system freezes up without warning. The only thing I can do is hold down the power button until the system turns off. I've not been forced to simulate a power outage on the laptop (by removing the AC adapter & the laptop's battery) to turn it off.

My working theory is that the issue lies somewhere on the motherboard based on the different points where the freezes occur. I can eliminate the laptop's RAM & hard drive from the suspects since I have swapped components from another salvaged laptop of a similar model. One theory, which I'm resisting due to the high improbability, is that the laptop suffers from overheating. The main reason I'm resisting this theory is that the entire system is resting at room temperature (as mentioned above) & it shouldn't be overheating within the first 15 seconds (being generous with here) of turning the system on. When the system was working (prior to this issue), the system usually rested on a hard surface & the back of the laptop was occasionally elevated to help with ventilation (& to help eliminate lighly roasting the user's leg(s)).

I do have a working knowledge of the system's build that I can tear it apart to some degree (can access to RAM, WiFi Card, Hard Drive, Keyboard, processor & heat pump with a little work) & I'm not willing to cough up the $60 or more for professional diagnostics & repairs as it isn't really worth the cost compared to the value of the laptop... so if I can't fix it myself or cheaply, I'll likly give up scrap the system (or at the very least, salvage it for parts in a future project).

HERE'S WHAT I'M LOOKING FOR IN AN ANSWER:

1) What is the probable issue with the laptop? -- Please give a reasonable explaination for the cause of the issue

2) What repair options would you recommend (IF ANY)? -- Please be aware that I'm not planning on spending much (if any) on parts & I'm not planning on tearing the entire laptop apart either. DO NOT RECOMMEND BUYING A NEW LAPTOP (as only spammers will do so)!!!

Thanks in advance for any assistance rendered to my question, even if it just confirms my suspicions. Hopefully somebody has a reasonable answer (even if it's something I may not like) to help me out!