Faith (as defined in Hebrews 11:1) is not so much belief, as the ability to see things that cannot be seen through human perception, alone.  As believers, "we walk by faith, not by sight." (2 Cor. 5:7) Of course 'seeing is believing' as they say–so it is connected to belief.
 
To be justified is to be made just–to be made righteous.  I think Paul is saying that we are not made righteous by what we do but by our faith, the way we see things.  The way we see things effects not just what we do, but our very being.  It changes who we are.  I could do all the good works in the world and if I was still a cold hearted sinner, inside, "if I have not love, it profits me nothing". (1 Cor. 13:3) But, as we learn to see things as they are we become humble, loving and grateful beings…and from this love springs the good works to which we have been called, and the willingness to take up our cross.
 
The question of whether we carry our cross in order to be justified or because we are justified is sort of a "which comes first, the chicken or the egg?" deal—faith, from which comes the ability to become righteous, to be made just, is given to us by God but it is manifested in works that produce more faith.  Taking up the cross is both a result of faith and justification and also produces it.  I strive to be kind and I fast* because I have been given the eyes to see that these are the right things to do.  But in being kind and fasting, I am given even more faith, can see even more clearly, that these are the right things to do. 
 
*Being kind is a manifestation of love.  Fasting, is a form of taking up one's cross–a means to an end.  The end, of course, being genuine love for God and for one another.

In some ways I am a poor candidate to give anything near a definitive answer. I preach to teach and inspire and equip my flock to live the Christian life. The nuances you mention are way over almost everyone’s head. I think you can think of the group which would understand this in our church.

I have a little saying that “we are not saved by our works, and we will not be saved with works” (or something like that – it probably changes a bit now and then).

Of course, we are saved by grace, with which we must cooperate. This cooperation is the taking up of our cross, following the way of Christ, to know Him by experience. So I would not agree a statement like: “we justified because we take up the cross”, however, we will not be justified if we DO NOT take up our cross.

I read St Paul in this way. Nothing we can do makes us deserve mercy, but we must “do” to obtain mercy. We love him because he first loved us. We react to God’s mercy by changing. Without the effort to change, we cannot be saved. I think this effort involves the way of the cross.

I think St Paul was referring to all works. He was not specifically referring to ceremonial washings, etc. Nobody can fulfill the law, since if he is guilty of it on one place, he is guilty of it all (a poor paraphrasement, but I am sure you get the reference).

I am not a systematic theologian. I am just trying to take up my cross and encourage and help others to take up theirs. Because of my position in life as a pastor, I have little cause to delve deeply into this complex theology, since there are always those present who can only digest milk. As I said before, I am a poor candidate to tie all this up in any kind of systematic order.

I just know that I do not love God enough, and neither does my flock, and this is our main problem.