Yes, perfectly normal for it to start in second gear. If you want to or need to start in first gear then you'll either need to shift down to L or have a functioning kick down switch and activate it shortly after starting off.

Sounds like you need to Check/Adjust your pressure control cable aka bowden cable

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Current fleet
2006 E320 CDI
1992 300D - 5speed manual swapped

former members
1984 300D "Blues Mobile"

1978 300CD "El Toro"