I see the key to the problem as:
"it works fine when the engine is cold"
What is it that when warm prevents it from starting in first with the selector in 2nd?
'03 E320 Wagon-Sold
'95 E320 Wagon-Went to Ex
'93 190E 2.6-Wrecked
'91 300E-Went to Ex
'65 911 Coupe (#302580)