This is what I understand.
The French Foreign Ministry said [X], without giving a direction or a date.
This X (what the French Foreign Ministry said) is:
radar echoes from a satellite put the new debris finding about 2,300 km (1,430 miles) from Perth
The subject of the clause is "radar echoes from a satellite", the verb is "put", and the object is "the new debris finding". To summarize, it (the radar echoes) put the finding somewhere. This "somewhere" is "2,300 km (1,430 miles) from Perth".
With that, it's now easier to discuss your concerns.
Are "debris" and "finding" two separate words or one word?
The phrase "the new debris finding" is a noun phrase. It's a "finding". A "finding" means a result of an investigation. But what is this finding about? It's a finding about the "debris", so we can call it "a debris finding". And because it's the new finding, the noun phrase has become "the new debris finding". In short, you can think of them as two words ("finding" is the headword of the noun phrase, being modified by "debris", which is a noun acting as an adjective), but they are part of one noun phrase--"the new debris finding".
Is the word "echoes" a verb or a noun? (My intuition is that is a noun.)
This "echoes" is a noun. This noun is "radar echoes". They are "radar echoes from a satellite".
What does the so called simple word "put" here in this context really mean?
This "put" is a verb (in the past tense). It means "to place something somewhere". In our case, the "radar echoes" placed "the finding" "2,300 km from Perth".