I am sometimes too blunt, so please don't take this the wrong way. I
think what the other posters are saying is that for an inexperienced
backhoe operater to work on the deep trench when you have saturated
ground may be a bit dangerous. Sorry to just blurt it out like that,
but I think they were saying that for your own good. Even experienced
operators have been known to land in the hole, on occasion.

If it is surface water, and not some huge spring under your house/yard,
4 feet or so should work fine. I think that most people also feel that
a trencher is safer because the cut is narrower. If, hypothetically,
you are digging and a dog goes in the ditch, you can't get it out
without endangering a person in such a narrow/deep cut. On a
commercial site, you would put a steel slip form into the trench to
keep workers safe.

If you are going to do this, try a trencher. Also, forget about the
plastic. Fill the ditch with crushed stone. That'll keep the water
moving and will be zero maintenance. Granted, it may eventually fill
in, but you won't own the house be then. But, put some landscaping
cloth on top to keep the topsoil from going into it. This would be a
simple and safe way to do it.

If you have clay soil, just go to the level of the clay.

If you start from downhill side and work up, it shouldn't fill up with
water -- but you will get more sediment running out and you should
control that (put in a bale a hay to slow the run-off.).

As for the sewer pipe. In all likelihood, you will break it. Almost
anyone would. So go to your local builder supply store and ask how to
fix it. Then get what you need before you start. You can always
return it.

If it were I, I would also rent a big-ol snake and snake out the line
since you have it open. Then it will be good and clear and won't cause
you any trouble in the future. After all, it goes under YOUR house.

I did trenches on both sides of my house and across the back about 3
years ago. But I didn't have to go too deep. I hired a couple of
local kids who were on the football team and had them dig it by hand.
Let's just say, by the time they were done, they would looking forward
to football practice and conditioning because it was so much easier!
(The kids bulked up pretty good and had a great football season)