How close to the house is the grass?

They sell seed blankets at HD and Lowes. I imagine those would work even better than netting in your situation.

Here is a link that might be useful: seed blanket

if they're walking directly on it during germination and after the seed sprouts, there is nothing that is going to save it. If you use a perennial rye that will come up in about 5 days, then keep them off it for a little while then it might have a chance.

How close to the house is the grass?

I'm not sure why this matters, but it gets as close as maybe 2 feet from the house.

If you use a perennial rye that will come up in about 5 days, then keep them off it for a little while then it might have a chance

It's tall fescue. Where I live it typically takes 10-14 days to germinate. There's now way I'm planting rye. I think the longer it takes to germinate the better because they'll have more work done prior to germination.

I'm think of building some walkways with some 16' lumber. Maybe place a few 2x4 supports under so I would have a few dead rectangles as opposed to a dead path.

Good luck getting them to use the walkways!

Have them build a temporary deck type walkway. Build it on 4x4s for their safety and stability. Make it about 4 feet across minimum.

Sheetrockers, or more specifically mudders (those who mud & tape) are notorious for cleaning their tools and buckets with a spray hose in your backyard. Makes a freekin mess. Most of them don't seem to care and will try to get away with it if they can.