Need to fill deep gap between ceramic floor tiles to be Grouted?

Two options. A wet, loose mix of thinset mortar in a grout bag and squirt it into the crack(s) or let the grout fill the crack. Rope, sand or foam will not support your grout line and ultimately fail.

I've been a contractor for 45 + years and have installed thousands of sq. ft. of tile, yet I am a bit confused.

I do agree with answer one however.

Is the two inch a transition area, as if a threshold? Is it a 2" drop/rise? That's not only a safety hazard, but would have been worth it to craft a sub floor before tiling. Too late now of course. anything wider than 1/4 in. grout lines, most often in commercial sets, sound be non sanded grout.

Is the foundation now, the offset, the quick-set all secure at this point?

If anything, this sounds like a situation will have to be maintained long term. In this case, I'd probably try the grout mixed with quick-set to fill any under tile spaces.

No offense to you, but this is a major segment to my business. Called out to fix.

When grouting use a Sanded grout. The only problem you may run into down the road is since you have an uneven subfloor your ceramic tiles may crack down the road where it is unlevel. I suggest going to get an extra case of the tile that you put down so in the case of this happening you can lay down another tile in it's place because wherever you bought it you may clearance or discontinue the tile and then you'll be replacing the whole thing.

You will probably have to scrape the stain with a sharp object and remove the grout until the stain is removed . If the stain has penetrated the grout deep enough you may have to re grout the spot with new

there are floor leveling compounds on the market you can buy. It is self leveling to a point. You need to help spread it around but it levels itself before it dries.