Traveling to Europe with US refugee travel document

I will use my US refugee travel document and need to know whether it is machine readable?

At the bottom of the photo page of a machine-readable passport or travel document, there are two lines of coded gibberish including upper-case letters, numbers, and <<<< characters, as in this example:

You would be able to recognize your name in the first line and your date-of-birth and the document's expiry date somewhere in the second line. These lines are what the machine actually reads.

When it's time to check in for my flight, will I be able to use the Delta kiosk at the airport to scan my passport to get my boarding pass. If I go to the check in counter, will they enter the passport information manually, or can they scan it?

Probably the kiosk will scan it fine. Otherwise the check-in desk certainly can.

What else would I need to take, just in case, to show the airlines and passport control in Germany that I can enter Germany visa free for up to 90 days?

See the question that Zach linked to.

Once they stamp my passport in Germany, will I be able to go by train to Belgium and the Netherlands. If the authorities stop the train by the border and check my document, will I be alright? What about going to other countries, like Spain or Italy; would I have issues flying there if I am already inside the Schengen area?

According to the linked question above, Belgium and the Netherlands would be OK.

Neither Spain nor Italy allows visa-free travel on US refugee travel documents (this is a question that is not harmonized at the EU level), so even though you may be able to go there without being stopped at the border/airport, you wouldn't be lawfully present and could get into trouble if local police took an interest in you for any reason. Which kind of trouble is unclear; it could range from a fine to being detained and subsequently banned from the entire Schengen area.

Whether you could fly to Spain or Italy depends on the airline. Ryanair, for example, is (in)famous for insisting on verifying the visa status on foreigners on Schengen-internal flights; some other airlines take a more laissez-faire approach, as long as you document you're the person a ticket was bought for.