Car starts and shuts down after few seconds?

Was it fine before you did all the tune up? If yes, you could have just missed a vacuum connection somewhere, or it may be loose. Small vacuum hoses are easy to overlook.

It was ok. Changed alternator then engine light came on. Wife had a few (her words) coils changed but didn't know which ones. I changed all coils and plugs and pcv valve. I plan to take it apart (except coils and plugs), check everything over and try again.

I don't know Lex very well. BUT I don't see why it wouldn't have a fuel relay. You get start right, but it doesn't stay so that's a sign of a disturbed circuit ( or relay old/ bad). It should be in fuse box, but again, I don't know Lex too well. good luck

All hose are connected and snug. Found wire broken off but don't know where it goes.

does it reach the area of the grey plug in the middle left of the photo?

Are you sure it is a wire? It looks like a black vacuum hose to me.

I faced the same problem after changing ignition coil The problem was because I tried to start the car without fixing the part under the wipers

I've had the same problem with my 2008 Lexus Rx 350. Start, go dead and start right back up. Not getting fuel. None of my mechanics could figure it out..... Try this. Turn key on for 2 seconds and start. Mine starts every time. Hope this helps you also.