What causes codes P0705 P0706 P0410 P0404 P0301?

You have a lot going on there multiple problems.The code P0301 is cylinder 1 missfire code.The P0705 and P0706 are for the selector switch you will need a multimeter to diagnose that problem.The code P0404 is egr desired position open performance meaning the egr valve could be sticking open.The P0410 code is for the injection having a fault that will probably be the hardest code to diagnose those are usually most likely a bad combo valve bad wire to the air injection solenoid which i have seen two cars i worked on in a row with that issue.Clogged air injection ports in the head and a bad air injection pump.That problem has no drivabilty issues the high idle concern how high does it idle rpm wise?The first code i would work on would be the selector switch codes.Those single cam 2000-2002 S model saturns are known for intake leaks around the number 1 cylinder so that could explain a high idle and the code P0301.

Saturntech9
Thank you for the reply!
Have a couple of followup questions.
1.The guy I bought it from downplayed the problem and said it was drivable. I did drive away. But, will it make the problem worse if I continue to drive it?
2. Is the switch a costly fix? I'm not real mechanically competent.

With the transmission and slamming in forward and reverse that could do some damage.That switch could also cause slipping and slipping could damage the clutches.The switch is about 50 for the part.Labor is about 1Hr and say 1hr to diagnose it.I hope you got a good deal on the car especialy since you said you cant fix the car yourself.

If it could be fixe d by a regular person then it's not necessary to have the dealer do it?

If you take it to a independent shop make sure its a good one.I have seen a lot of people struggle to diagnose those air injection systems/codes.For me it wouldnt be a issue but iam a factory trained saturn tech.You dont live in california do you?

Followup question -
Repair shop says the valve gasket needs replacement. Cost $600.
The transmission selector switch replacement. Cost $310.

Said the gasket should be replaced first because that could be causing the tranny problem. Don't have the printout showing which came first P0705/6 or the P0410(?).

How likely could the problem be caused by a bad gasket? i.e. which would you do first?

So they want 600 just to replace the valve cover gasket?If so run away from that shop thats is way too high to replace that simple gasket.The selector switch is only like maybe 60 and about a Hr and some change labor hrs to change so 310 i think is way too high.The valve cove gasket is a Hr and some change at most to replace plus like 22 for the part.I dont see how the valve cover is causing the selector switch codes.Are you handy with tools?If so lets change the valve cover gasket its pretty simple to do and the selector switch isnt that hard to change ethier.For the selector switch you need a 13mm deep socket with ratchet a big flat blade screw driver and a multimeter that reads ohms.For the valve cover you need a small pockect screw driver 10mm wrench T-40 torx socket with ratchet and some brake cleaner to clean the oil up.I would also replace the silcone molded gasket that is a silcone applied gasket with the felpro perma dry molded rubber gasket.I have used them on a couple of those and they work great easy to install and dont leak.

saturntech9
You've been a wealth of information! Thank you!

The "Donate" button underneath the text box, does that go to you or someone else?

That would go to me if you donated if your donating thank you in advance it really helps.So were you thinking about replacing those parts yourself?