MacPro 4,1 starting with grey screen after SMC and PRAM resets

these items have no place on your high-quality Macintosh computer:

User Login Items:

    [Not Loaded] AppCleaner SmartDelete (Julien Ramseier - installed 2019-02-25)

        Modern Login Item

        /Applications/AppCleaner.app/Contents/Library/LoginItems/AppCleaner SmartDelete.app

Launch Agents:

    [Not Loaded] com.avast.hub.plist (? 0 - installed )

Effective defenses against malware and ot… - Apple Community

How to download and install macOS - Apple Support

up through Sierra can be downloaded and unpacked on a Mac, and the installer is placed in the /Applications folder that can be used to make a USB-stick installer

for High Sierra, you need to use the link in the article and download through the Mac App Store, but the computer doing the download must be capable for running that version of macOS.

Create a bootable installer for macOS - Apple Support

if you name your USB-stick exactly MyVolume, those goofy long scripts can be copied from the article and Pasted directly to the Terminal command-line.

To do a restore on to a drive, best practice is to have MacOS installed and stable on that drive already, then use Migration Assistant to bring your files over from time machine backup driveMove content to a new Mac - Apple support

Transfer to a new Mac with Migration Assistant - Apple Support

Migration Assistant 'takes over' everything, and takes a surprisingly long elapsed time. First it may need to compute a Spotlight index of the data. Once data transfer begins, it takes a bit longer than a FULL backup, likely all afternoon to overnight. You may want to set this up late in the day and let it run overnight, and be ready for it not to be done by morning. 

"the best way" is to use your Time machine backup from the old Mac as the source for Migration Assistant running on the new Mac. Connecting that drive using USB-2 is as fast as almost every Rotating Magnetic drive, and will not produce a noticeable slowdown doing this transfer.

Once you have MacOS up and running reasonably, the method you use to get different versions changes dramatically. You use the running MacOS 10.11 El captain to download 10.12 Sierra.

To move to 10.13 High Sierra, you switch to Mac App Store download, But only by using the links in that same article:

How to download and install macOS - Apple Support

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your boot drive is Full. You may be moments away from a very spectacular crash.

you have only 26.23 GB free, and the swap file is on that drive, so just open a few more web pages and kaboom! For reference, MacOS consumes over 9GB of drive space going from a cold start to fully operational.

in addition, you have not enabled TRIM, so that drive has slowed to the speed of a rotating magnetic drive:

Performance:

    System Load: 3.86 (1 min ago) 4.73 (5 min ago) 2.75 (15 min ago)

    Nominal I/O speed: 0.03 MB/s

    File system: 15.16 seconds

    Write speed: 233 MB/s

    Read speed: 269 MB/s

use Terminal command:

sudo trimforce enable

and enter your password (which will not be echoed) and follow the directions, culminating with an automatic restart to engage.

https://www.lifewire.com/enable-trim-for-ssd-in-os-x-yosemite-2260789

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The code supporting option key boot is all in ROM.

if you can't get a gray screen, your Mac has hardware problems.

The code that runs Recovery on that Mac is on a drive -- it does not have a ROM version of Recovery.

You should also get a 'wait' cursor, but that does not always happen. Then it discovers possible bootable drives over the next several minutes and adds an icon for each potentially bootable drive it finds. When it has scanned through everything once, you usually get an Ordinary pointer cursor.

Is the display actually showing a picture?

my 30-in cinema display recently conked out, and it made me nuts figuring it out. mine turned out to be the Power supply block, and the telling symptom was that the ON light (lower right corner) was not lit when the display was dark, but nominally ON.

that sounds good for the display itself.

what about the graphics card?

I have had an interface (especially the higher-power DVI or its cousin HDMI) blow out, and got no display after that. I had to switch to a Mini DisplayPort adapter (to single-link DVI) to get a 1920 by 1080 picture, showing the card was good, but the interface was bad.

Do you have the diagnostic, originally on the original release disc-2?

Do you have another external or internal drive, on which you could install a macOS and boot from? or the ordinal Installer DVD, just to see if you get a picture and show more stuff is working?

Another way to boot to an alternate boot drive is, in the running system,

system preferences > startup disk

and specify a different connected drive as the startup drive. Then choose Restart from there.

As always, the startup disk is a parameter in the parameter RAM, and is reset by PRAM reset.