Did Luther add a word to Romans 3?

Yes.

When people translate from one language to another words are sometimes added and sometimes removed (for instances, Matthew 6:1). One can compare any two Bibles and find thousands of words to argue about. Citing Morris Protoctor

Sometimes Bible translators, for various reasons, will both insert extra English words and not translate all Hebrew and Greek words.

I think Catholic opposition to Luther's choice of words, and obsession with this one in particular, indicates what the Protestant reformation was all about.

Yes, clearly and obviously, Luther added the word "alone" to his translation. But that's not the real question to ask. If you'll permit a small digression, the real question is why. It is clear that his reason is to make the meaning of the greek in the receptor language (german). In other words, the greek carries with it the nuance of 'alone' in the greek itself without the word, 'alone.' (“⸀λογιζόμεθα ⸁γὰρ δικαιοῦσθαι ⸂πίστει ἄνθρωπον⸃ χωρὶς ἔργων νόμου.” (Romans 3:28 NA28-T)) So, for the sake of bringing this clearly over into his target language he adds the word alone. Here's an excerpt from Luther's open letter on translating

"You tell me, besides, that the papists are making a tremendous fuss, because the word sola is not in Paul's text, and this addition of mine to the words of God is not to be tolerated...But because I knew—and still see with my own eyes—that none of them knows how to translate, or to speak German, I spared them and myself that trouble.."

To word it slightly differently, the issue to Luther was a translation issue far more than a doctrinal issue in his translation of Romans 3:28. He was simply striving to bring the precision of the greek into the German. Paul is setting up an exclusive statement. "We conclude that a person is declared righteous" (λογιζόμεθα ⸁γὰρ δικαιοῦσθαι). But then Paul front-shifts the instrumental dative for emphasis (prolepsis) out of its normal position. So we would add weight and emphasis to this by saying something like "We conclude that a person is declared righteous by faith!!!" Next Paul continues with a very strong ablative adverb (χωρὶς). We would express this in english by saying something like "without any sort of law-works at all!"

The arbiter/moderator is welcome to strip away this comment. But I added it initially for a reason. The question is incomplete. It's a question that betrays a lack of understanding even the basics of translation practice. For when you say that Luther added a word to the German it carries with it the unspoken assumption (accusation) that he added to the meaning of the Greek. That is simply not the case.

Let me give you an example: In my translation class on the Vulgate we kept running across the phrase "it was a long time until his nose became hot." We had absolutely no clue as to what that phrase meant. It was only later, when I actually took Hebrew class that I realized that the phrase was the Hebrew idiom for taking a long time till someone "blows their top", losing all patience. So, to translate that from the initial language (Hebrew) into the receptor language (english) you need to change the english wording so that you do not change the Hebrew content. It is the same here with Luther's wording of this passage in Romans. This is basic, elementary linguistic practice, not ax-grinding, agenda-ridden, doctrine-shoe-horning habit.

Also, it may surprise you to learn that Greek is not english. (-: Greek can use an exclusive particle like μονον to express an "alone" thought. But it does not have to.

As I mentioned, it is fine with me to strike my big long comment. But the question is incomplete as it is written. I don't think that the questioner meant it that way. But, nonetheless, that is how it is. Without a basic working understanding of translation theory it is easy to conclude that Luther was adding to God's word.

Any question about James should be asked separately and not in this thread.