On the rough idle - does our VRS have a vacum assisted line to a brake booster on the brake master cylinder?
I would look into that for the rough idle (air leaks)
I had a similar problem to point 2 - was just the exhaust joint loose under the gear lever - once it got hot expansion made a tight seal
2012.1 Skoda Octavia VRS DSG Wagon - Carbonio cold air intake and pipe - HPA Motorsports BBK 355mm rotors 6 pot calipers
APR Stage II ECU - APR 3" exhaust down pipe & high flow catalyst
APR/HP Roll bars - Eibach springs and Bilstien shocks
Supaloy lower control arms - Enkei 18*8 Wheels
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