Mac Pro running OS El Capitan refuses to complete startup

When I suggested above that you NOT replace like-for-like I wanted to be sure you did not simply replace that old 2600 card with a fixed up version of the same old card, because graphics technology has moved along briskly since the 2600 was a good card.

You can still buy a good Apple-firmware 5770 card for not a lot of money, and it will work better and run much cooler than the original card, because the chips on it are newer technology that runs cooler.

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Aux câbles sometimes come with the new card, but are always re-usable from the old card. they are about 6 inches long and plug into some sockets on the backplane nearby, because the PCIe slots can't provide all the power needed to run most graphics cards.

DVI Single link does not use the six center pins (whether present or not) and supports display up to 1920 wide, but not wider.

To simplify, I suggest you cut the disk drive and software out of the equation.

The simplest thing that Mac Pro can do is boot to Startup Manager, whose code is all in ROM. Invoke with the Alt-Option key held at startup.

it will draw a gray screen, then proceed to add ani Icon for each discovered potentially-bootable drive. the picture should stay stable and predictable.

Alt/Option boot is VERY simple, and its code is all in ROM.

if you don't get a stable screen, the fault is Likely in the graphics card.

Perhaps the card you received was not as good as the seller expected. Sometimes one interface on a card can collapse while other are still working. The seller may not have tested all interfaces. That card may work GREAT with its mini DisplayPort interfaces, but the DVI might not be working.

<<. I have been given some advice that I should keep this machine 'away' from the internet - that is not allow it to access my WiFi. >>

There are two kinds of vulnerabilities that have no way to be resolved with that setup.

The first is an obscure attack vector that could expose prefetched cached data to break into other processes running on your computer. Its use requires an attacker with state-level sophistication, but it is a vulnerability nonetheless. Intel declined to provide adequate firmware revisions to cut it off, saying these processors were too old. Apple revised Safari to blunt most of its use without crippling multi-processing performance.

The second is more mundane software updates that cut off newly-discovered vulnerabilities in macOS. Once 10.14 Mojave was released, any new security updates were not sent out for 10.11 El Capitan. So your older MacOS may not be able to defend itself against current attacks.

The overall threat limit may not be that high, but it is definitely not as low as many users would like. If you are a political activist or official of any kind, or have any comparable visibility, you should NOT connect this Mac to the Internet.

(The reverse is deliberately left hanging. It is up to you.)

exactly which model Mac Pro?

 menu > About this Mac

(please don't post your serial number, it attracts vermin.)

do you get ANY activity on the display at all, such as gray [not completely black] screen?

what cables adapters and docks are you using to make the connection to what display?

Believe in yourself. You are on the right track.

Readers here are all glad you got it sorted out.

YOU did the heavy lifting, and you gained some increased confidence in your ability to "Make it go!"

The initial "chime" sound is generated in software when your Mac passes the Power-On Self Test. If the chime occurs and/or startup continues, your Mac is working. The screen should light up and draw a blank gray screen. Then on to the Disk Drive.

Accessing the Boot drive:

The solid Apple is not in the Mac's ROM at Cold start. The Apple logo can only appear when it is fetched in the first "blob" of software loaded from a 'magic' place on the boot drive, or re-run after a Restart. Then a whole lot of stuff is initialized, and the progress Bar moves part way across. After a cold start, seeing the solid Apple appear says your drive was able to produce the software that contains the Apple logo.

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the trash on the screen at the same time suggests your graphics card may be failing. if this is the original card, I recommend you not replace like-for-like, as there are much better alternatives available.