XPS 8300, starts for a few seconds then shuts for a few seconds

There is no way to check the CPU other than installing it in a motherboard. It is more likely that you could have damaged the socket than the CPU. Electrostatic discharge by touching the electrical pads on the bottom of the CPU could damage it. The protruding pins on the CPU socket can be easily bent. It is called a LGA 1155 socket because it has 1155 pins. The CPU has notches so there is only one way to install it properly in the socket.

Okay so I just ran the paper clip test and the fan on the power supply runs constantly. I couldn't see it spinning because of the housing but I could hear it faintly and I could feel some air blowing out of it.

Do you have a multimeter that you can use to check the voltages? I believe the power supply fan runs on 12 volts. With a multimeter you can check the 5V, -12v, and 3.3v. 

Did you hear/see the CPU fan and hard drive fan spin up during the paperclip test?

We won't see your image until the moderators approve it which might take a few hours or a day...

And be prudent. If you don't know how to use a voltmeter, Don't do it or you could risk further damage....

Something like this? I don't have one or know how to use it to test the power supply voltage.

That is not possible. The instructions said to unplug everything that uses power from the motherboard. Even if the CPU and hard drive fan were plugged in they wouldn't be getting power to them because I had to unplug the pin connector from the power socket on the motherboard so I could stick the paperclip into the pin connector.

Not quite true. If the CPU, CPU fan, case fan and HDD are still connected to the motherboard, they should spin up. It just won't boot when you do the paperclip test.

Okay, so I plugged in one of the power cables to power2 on the motherboard, connected the power cable to the hard drive, plugged the CPU fan and case fan back into the motherboard. The power supply fan came on and I heard the hard disk spin up for the first time since 2012!! The CPU and case fan didn't come on I guess because no power is going to the motherboard. I noticed there might be a problem with the main power connector. One of the pin holes is empty. Is that normal?

Yes, it is normal for that particular pin to be missing. As for the mutimeter, the picture you posted is one of a muitimeter, but as Ron said if you don't know how to use one or have one. you should not try it. At this point I can't think of more testing you can do with the power supply and since the power supply seems to work, it appears that you have a bad motherboard. You did indicate that you had remove and re-installed the CPU so I hope that is installed correctly and is not an issue.

pin 20 is N.C. for years now.  minus -5vdc is gone since. ATX v1.3 and up it is gone now for ever.

since April 2003  GONE forever.

the bench test is here, (its just and unloaded test, we have power loads in our shop to do the real test)

but a bench test is valid and sure if that cable is pulled all things in the PC are now dead, no DC power inside the PC if 24 pin is pulled and 4 pin CPU power is pulled, (a captain obvious statement.)

if the PSU is ok and you stripped the mobo down, and it still is dead, that means the MOBO is bad or you bent any of 1155 pins in the CPU socket, and gee seen vast of those, with folks playing there an not knowing to leave the CPU alone.  (intel dot com) has  nice long and video rich support  page on now to do it right and is not trivial this. just the slightest slip of hand and bam mobo is wrecked, yah that easy. (no coffee drank before, LOL)

My guess is mobo has 2 faults,  one before you bent pins. now 2,

learn this about PCs first.

1. the CPU can and will shut down if over 90c. (or near) the magic if intel INSIDE>!
2. the PSU can shut down 3 ways, (that is correct) 3 ways, it fails PSU-inside, or sees any rail shorts or is told by the MOBO VRMs that the MOBO VRM are overloaded,  all in under 1 second flat.  ATX spec, that.

There is only 2 ways to fix a PC like that, the full PSU bench test, and a full MOBO strip down, leave the CPU alone.

to test voltage make sure the DMM is set to VOLTS not amps. AND THE LEADS ON VOLTS PINS not AMPS.

THEN CONNECT BLACK LEAD MINUS TO BLACK GROUND PINS ON PSU (step1)

then connect red test lead to ,  each power pin (one at a time),  3v, 12v, 5v, and make sure they are good. the first link above shows the spec on power, accuracy, and even ripple. (we scope ours in a shop for ripple and bad caps ,them bad, causes 40khz ripple huge, and most techs new are clueless on that. and why. SMPS technology here. is why.

if you google this , zillion hits, "how to use a voltmeter" or DMM , endless,

just do not slip and short pins to  other pins or grounds or casings. oK>>?

but the jumper clip test is what you do. that turns it on. seen in link 1 above.

My PSU can be vastly newer than yours  has more protections. the over current protect is first.

 OCP, SCP, OVP, UVP, OPP

this magic of protection saves folks for blowing up $500 CPU and $500 GPU cards, and more. 

POWER IS FIRST, in electronics, clean , accurate, safe and protected or bad can and will happen

Many and old PSU,. have bad caps, they ran for years ran hot and backed there guts out (guts = electrolyte)

and now the PSU is a giant noise maker not a PSU, at all.  RIP old PSU,  (why not upgrade it first)?

yours ran for  full decade,  I wonder if Caps run long and hard from 2009 are still good to day, (hummmm)

your mobo sells for $20 used and good. (no CPU use yours and put it in and not bend the pins, ask or ask intel.

a good PSU is only $50

for $70,  I can get a real nice DELL 3020MT,  Haswell i5 and all, and a better future. used and good.

go from your gen2 CPU to GEN4 or more, yours is Core i7 2600 3.4 GH

nothing wrong with that but PCs are cheap used, and a far far better PC cost less used than fixing gen 2 PCs.

a fact. of life. (econ 101)

If  the PSU powers up the HDD that's a good sign.

But as @Vic384 said, if you removed and reinstalled the CPU, so are you sure that's not the problem? And did you clean the top of the CPU and bottom of the heat sink and apply a thin, fresh coat of decent quality thermal paste?

If you didn't apply thermal paste, the CPU could be rapidly overheating and causing it to crash...

The problem I have with the computer turning on and off constantly is the same problem I had long before I removed the CPU. If I decide to buy another motherboard then I would clean off the CPU and CPU fan of the old thermal paste and apply new.

My computer did not run for a full decade. Not even close. I bought it new from dell.ca in 2011 for over $900 and in 2012 it would no longer boot up. It had been collecting dust ever since. I decided recently to see if I could find out what the problem was through the Dell Community. I think the consensus is that it's the motherboard. I'm tempted to buy a used motherboard to see if I can get it running again but there could be something else wrong with it or I could get a used motherboard that is DOA. The CPU fan did run during the paperclip test plus the hard drive was spinning when I connected power to it so that's a good sign.

What's the intended use of this PC, assuming you get it to boot, aside from viewing/recovering files on the HDD?  And if you haven't seen those files since 2012, do you really still need them now?

I doubt the XPS 8300 can run Win 10. Support for XP ended eons ago and, even if it was running or can run Win 7, support for Win 7 ends next week, so are you willing to take all the extra precautions to protect it from a hack attack?

Would the $ spent on a replacement motherboard (with questionable history and longevity) be better spent on a cable or HDD enclosure to house that drive so you can retrieve your files? If you put it in a HDD enclosure, you could use it as an external drive that connects to any PC via USB for backups and/or extra storage, giving your investment a lasting benefit. It's probably a SATA1 drive so make sure any enclosure is fully backward-compatible all the way back to SATA1. Some enclosures may only be backward compatible with SATA2 drives

If you replace the motherboard and it turns out to be the CPU itself, then what...?

Only you can make this decision. We just get to ask annoying questions...