Congradulations on your purchase and Welcome to the Greatest board on Earth.

Regarding your questions:

1. When you say "slow", relative to another diesel or you are comparing with a gas car. I am no diesel expert but 300D are 5 cylender cars and are slow at start up.
2. The rumble under the center console sound like a "center support bearing" for differential shaft.

This by no means an expert's opinion, we have very expert senior members who will, I am sure, share their opinion about this. There is also the diesel forum of this board and a lot of wealth of knowledge can be found there. I will also suggest you to use the search on these forum, most questions are asked before and will help you get your answer quicker.
I hope it helps.

Thanks

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87 420 SEL
Looking for a nice mid 80's MB diesel.
87 BMW 528e (Sold)
99 TOYOTA 4Runner
1989 Honda Accord (Sold)
1988 Honda Accord (Total)