GPU failed, Repeat chiming, cleaned, revived, Loud pop, repeat chiming.

as long as you have a working macOS, you can upgrade. if you have 10.11 El Capitan or later, you can upgrade easily.

On models before about 2011/2 DO NOT ERASE your only working Boot drive, because Recovery may not be available if you do that. Use the links in this article to obtain a later version of MacOS of your choosing.

If you have a working macOS, start by using the 'Mac App Store' section. You MUST use the links in the article, because these versions have been "put away" to avoid having users download them by accident. Follow the provided steps faithfully, as they are a little more complex than expected.

How to download and install macOS - Apple Support

The debugging steps when you see some activity, but it won't boot are:

Strip the machine down.

Remove everything is a slot including the graphic cards and the drives, and go for a [single, well-behaved] CHIME.

Once you have the chime, build back up ONE item at a time.

When you add something that stops progress, that item is likely BAD.

**If not getting a chime at all, there is a more complex variant where you also remove the RAM DIMMS, and watch the power light for a "help-me' code.

<< amd radeon hd 6770 4gb. >>

no advice on that one, it does not appear to be common card and for a Mac Pro. Whatever card you choose, you MUST have a version with Mac Firmware, or you will need to apply some unsupported hacks (blind) before it will light up. Commonly available aftermarket flashed cards include this, but you MUST verify that card has Mac Boot screens support:

AMD 7950

AMD R9 280/280X

AMD RX 580

If you EVER want to run 10.14 Mojave, you will need a card with Metal support. The ones I listed can be flashed for Metal support.

You can keep using your existing disks as long as you want. You can even format a new Disk as HFS+ Extended if you prefer.

For BOOT disks, they will be formatted APFS, PROVIDED they are SSD disks, with an APFS container-disk in new 10.13 High Sierra and later. The format gets slightly more complicated by the addition of APFS Volume Groups in 10.15 Catalina and later.

Catalina requires App that are 64-bit ready, and all older Apps not up to that standard, and most Drivers for really old devices will fall away at that point. Your Mac Pro 5,1 is not officially supported for Catalina or later.

those are all readily available modest-priced used cards that will all work in 10.11 El Capitan, and my list was listed in order of lower first, proceeding to higher capability.

The absolute highest capability cards get crazy because their power requirement are higher than can be supplied with All the existing cables together. Hackers have devised a 'standard' modification to modify the chassis wiring harnesses to get more power. It's called 'Pixlas PSU Mod'. Of course It is NOT supported by Apple.

Typical work does not draw the maximum amount of power. So users get around this by using one six-pin cable directly, and the other, a six-to-eight-pin stunt cable that has ALL wires present the length of the cable (not just short jumpers at the connectors).

If that works for you as it has for me, then you should be in business -- PROVIDED the card was flashed to support MacOS Boot Screens and Metal support.

NB>> The symptom of a power overdraw is sudden uncontrolled power-down, precipitated by the power supply to save itself from damage.

<< On the Pulse there is 1, 8-pin dock. >>

Then yours may be different than mine. There are Many designs from a handful of different card-builders.

You will need to use a stunt cable that connects BOTH MacPro 6-pin power outlets to one eight pin, if that is what your card requires. Try to get one that has many full-length wires (and no short jumpers) for best electrical performance.

did the seller include such a cable?

Is this card sold as 'Mac Boot screens supported'?

the rule of thumb is that a nominal 75 W comes from the slot, and an additional nominal 75 W from each 6-pin aux port. Don't get too hung up on those as firm numbers.

The ACTUAL upper limit is determined by the gauge and number of wires in the aux cord cables and the total 12 Volts the power supply can produce.

This class of cards do not generally require you to tap into the power supply cables to get more power. Try it the way it is. If you overdraw the power supply, it will do an uncontrolled emergency power-down.

The most demanding cards available may require that modification. Hackers will try anything at their own risk and just for fun, including installing stupidly over-powered graphics cards in these older Mac Pro models. Hackers refer to a 'standard' way of obtaining enough power to do that as "Pixla's PSU mod". But that should NOT be needed for the card you have. And if you are adding such extra things that exceed what the power supply can provide, there are suggestions available online that use an additional power supply to provide even more power.

The 8-pin connecter is capable of accepting nearly double the power of the typical 6-in cable.

You want a Y-cord stunt cable that connects BOTH 6-pin power outlets inside the Mac and connects to the 8-pin power inlet on the card. Try to get one that has many full-length wires (and no short jumpers) for best electrical performance.

DON'T Build your own. I am confident that the places you already named can provide such a cable, ready-to-go, because so many have trod this same path before.