Replace the size 2032 CMOS battery.  A low battery can cause all sorts of bizarre problems, including boot loops.  If by chance not the cure (and it's cure boot loops before), at least it's ruled out.

If still not working, upon starting, immediately keep pressing F12.  Does it allow you to see the Boot Menu?  There's a Diagnostics option.  Run diagnostics and see if it tells you anything.  It will take a while.

I'm not sure if you're GPU card is bad or not.  So I'm not suggesting reinstalling it.  It would up to you if you want to risk it or not if we get PC working again.

You were right about removing the GPU.  2 & 4 indicates video problem.

I take it the 780 runs normally without the GPU card?

PSU might be on its way out.  From YouTube videos, I've seen PSU's as low as 240w power 75w cards.  Your card is 55w.  It is possible your 780 has the high efficiency 255w watt PSU, which may still be too weak to have the 55w card added to a tower.

In case anybody wonders, it's not bottleneck.

Graphic card and processor will work great together:

Intel Core2 Quad Q9550 (Clock speed at 100%) with AMD Radeon HD 7750 (Clock speed at 115%) x3 will produce only 0.3% of bottleneck. - PC Builds

I have solved the boot loop issue, when i installed the gpu i removed the battery so the bios configuration went back to default which was the problem: the default option for drives is RAID always, being the HDD an SATA it caused the issue.

Now respecting the 2 4 lights and the gpu, i can confirm that its working has its being used in an Optiplex 745, previously in an Optiplex 760 too. Another 780, of identical configuration, also showed the same issue, could this be because of the PSU?

When i try to boot both cpu and gpu fans spin, but when i presh the test button on the back of the PSU only the CPU fan spins, while the gpu tries but fails, does this means somethings?

Thank you. 

I have changed the PSU for an 500w one but still wont boot up with the 2 4 lights on.

To be observed: 

When first turned ON lights 1 3 4 will be on then turn off and ON twice.

After 2 4 light always, fans keep running, dvd drive can be opened, no keyboard light.

GPU works in an Optiplex 745 and worked previously in an 760.

Pc runs without the card.

Another 780 presented the same problem

Could you post photos of 2 labels? - One on the original PSU and the new 500w PSU.

Both PSU's may have something in common - not enough power on the 3.3v/5v rails.

It's not unusual.  I just got done helping somebody that put a 750w Enermax in their Precision T5500 which wasn't working.  The Enermax only had 100w on the 3.3v/5v rails as compared to Dell's 150w on a 750w watt PSU and 225w on a 875w PSU.  Point being it's not just total watts.  Dell's 305w PSU's also have 150w on those rails.  This thread might be worth a read noting speedstep's response and photo of 305w PSU label. - https://www.dell.com/community/Desktops-General-Read-Only/OptiPlex-780-MT-Power-Supply/td-p/5119091 

OldNew

Granted I still don't know if the original PSU in your 780 was 305w or 255w, I found an image of a 255w label to let me know how much power is on the 3.3v/5v rails.  If I read that right, 80w is way too underpowered for adding a GPU card.

The reason your card works in other Dell models is because they probably have higher watt PSU's than the 780.

Image will appear when approved by moderation:

Thanks for posting photos.  I can't see them yet.  They'll appear once approved by moderation.

"If I read that right, 80w is way too underpowered for adding a GPU card."  Not meant to be confused with GPU cards running on 12v.

While we're waiting for images to be approved, I noticed the 255w PSU has 13A & 7A on its +12v rails.  The 745* and 760 you mentioned, if they were MT's, likely had 305w PSU's.  The one for the 760 & 780 has 15A & 10A on its +12v rails.   As for a 305w PSU for a 755, it has 18A on its two +12v rails each.  If your 55w card is drawing off of the 7A rail, that could be where the problem lies.  12V x 7A = 49W.  That's 6W below what your card needs.  And if there's anything else on that rail...  *I couldn't find PSU specs for the 745.

YouTubers get away with running 75w cards on 240W SFF PSU's.  Those PSU's may only have 60W combined on the 3.3v/5v rails, but have 17A on the 12v rail.

One of our Rockstars is more adept with PSU's than I am and might chime in if he wants.  Either way, we'll get it figured out.

Depending on how much work you want to do on a 780, you might simply want to adapt a Dell 305w PSU from another Optiplex, like a 755.  305w PSU for 780 posts some lower numbers than the aforementioned.  Adapter for MB would be required - https://www.amazon.com/Replacement-Power-Supply-Cable-Optiplex/dp/B07GJ2CH8D/ref=mp_s_a_1_4?dchild=1&keywords=24-pin+to+mini+24-pin+adapter&qid=1592886397&sr=8-4 

The 305w PSU for the 760 posts the same numbers as the one for the 780, so that could be an option without needing an adapter.

Label for 305w PSU for Optiplex 755:

But going by that the new PSU should have worked has it has:

+3.3VDC  28A

+5VDC  30A

+12VDC 30A

-12VDC  0.8A

+5VSB  2.0A