"Esta mesa es de madera."

I do not understand "This table is wooded" ???

I seem to vaguely remember duolingo accepting, "This table is made of wood" here, but I might be wrong.

I completely agree.

No, it's not proper English. This table is made out of wood, there is no short form like wooden!!

Yes that is accepted because it is how it is commonly said. It is actually good to translate it that way as long as it reminds you not to try to put the "made of" in the Spanish. You could use a passive voice construction in Spanish but it would not be the common way to express it.

The from/of concept is interesting. It is true that from is always talking about origin, not just national origin. But I agree that when it comes to wooden tables that distinction is to fine to use. I would not even wonder whether I was speaking about the fact that someone took a piece of wood and fashioned it into the table, or just its current organic composition being of wood. It would not make a difference. But I have heard people say that the human body is made up from 60% water instead of of 60% water and found that strange. It might be just the statistics, but it might be that from makes me think of God standing in a laboratory concocting man.

As for wood table vs wooden table I would tend to agree that a wood table might be for wood and not made of wood. This is the type of sentence, however simple, that is challenging for Duo. When English has so many more ways of expressing an idea than Spanish, and the way that is the most direct translation is probably the least common it requires Duo to accept more options which are linguisticaly distinct. There is a obvious common expression in Spanish as there really isn't another way to say it in Spanish. But there are several ways to say it in English with no choice being obviously more common. But if it weren't for the existence of the word wooden, wood table would have to work. I can think of only two words modified in a way so as to mean made of, wooden and golden. If it were plastic, glass or metal those nouns would be used as is to modify table. Spanish does have a word for golden (Dorado), but wooden is de madera.

I believe "this table is from wood" would be translated as "esta mesa es de madera", wouldn't it?

For the same reason that "This wood table," is not the same as, "This table is wooden."

It's implied. "....es de ____" is translated idiomatically.

It's "made of" or "made from"

This table, not the table.

You missed the "this"

I agree

You cannot say of wood, is made out of wood., or a wooden table.

This is hardly a "really bad" translation. But the bottom line is that Duo's goal is to teach you Spanish, not how to translate. Translation is just a way to demonstrate your knowledge of Spanish. But like in Math class, simply getting to the correct ultimate meaning isn't always enough. Choosing a grammatically correct English answer which best reflects your knowledge of the structure grammar and syntax of a sentence is the goal. Of course "reflecting your knowledge" doesn't necessarily imply structuring the sentence similarly when there are standard difference between the languages, like using a possessive instead of a prepositional phrase. As a native English speaker with more than a few years on me, I have used the equivalent of ALL of the suggested translations when discussing the material something was made of. I have even used the less common form which is the literal translation of this sentence, This table is of wood. None of them are grammatically incorrect.

I think the major issue here is actually an English one, not a Spanish one. If we said the table was metal, plastic, or even a specific type of wood like mahogany or maple, it would be more parallel. But we still most commonly use the adjectival form from Old English wooden, instead of just using the noun form as an adjective as we do for most other nouns. It affects the structure we expect, although much of that is actually somewhat based on the individual or the particular region or subculture. I am curious, however, why you didn't include the translation This table is wood. While people still tend to still say wooden when the word is used as an adjective/modifier preceding another noun as in wooden tower, at least in my neck of the woods most people I hear would use wood as the predicate adjective. In that situation Duo also went a little off, but as I say it is an oddity in English involved that affects how different English speakers would structure this sentence. For a table made of Mahogany I see many fewer issues with all of the following structures. This table is Mahogany, This table is made of Mahogany. This is a Mahogany table. or even This table is of Mahogany. Interestingly, if someone chose to use the adjectival form metallic instead of metal to describe the table, I would be less certain if the table were actually completely constructed of metal as opposed to just painted in a metallic paint, but that's another issue altogether. I do believe that Duo used to accept This table is made of wood as an answer here. It is only problematic here to the extent that it might leave a struggling student confused trying to understand where the made came in. And, of course, Duo likes sentences with translations that work well to produce each other when translated. And while This table is wooden would cause me to actually think of the de madera translation, This table is made of wood I would probably translate to Esta mesa es hecho de madera. While simply saying de madera is probably the most common way to say it, hecho de madera is certainly common enough and quite grammatically correct.