Is your spouse's happiness your responsibility?

Prophets and apostles have counseled the members of the church a number of times that a married couple should each seek for the happiness of their spouse.

For example:
President Gordon B. Hinkley
True love is not so much a matter of romance as it is a matter of anxious concern for the well being of one’s companion. (link)

Elder Russel M Nelson
Harmony in marriage comes only when one esteems the welfare of his or her spouse among the highest of priorities. When that really happens, a celestial marriage becomes a reality, bringing great joy in this life and in the life to come. (link)

Elder Jeffery R. Holland
Find someone [to marry] ...who finds his or her happiness in your own. (link)

President Spencer W. Kimball
If one is forever seeking the interests, comforts, and happiness of the other, the love found in courtship and cemented in marriage will grow into mighty proportions.(link)

Does this mean that you are responsible for your spouse's happiness?  Are they responsible for yours?

In a way, yes you are...
The answer is 'Yes' in that we have a responsibility to do what we can to promote the happiness of our spouse, it is part of our stewardship as a spouse.  We should know (or find out) what our spouse needs to feel loved, happy and content with their life.  Not just with the marriage relationship, but in all areas of life.  If it is within our power to bring them greater happiness (without breaking commandments or laws of course), we should do it.

It may require us to make changes or sacrifices.  I may mean doing things you wouldn't normally do.  It may require gently helping your spouse overcome their challenges, or patiently waiting for the time they are ready to take that on.  It may just be giving them support and encouragement and expressing faith in them as they confront things you are not able to do anything about.  It can be giving them your best, honest counsel, forgiving them of their mistakes, and having compassion and charity toward them in their imperfections. It may mean seeking their forgiveness for things we've done.

Ideally it is supposed to be a two way street where your spouse does the same for you.  In practice that is not always the case, but even then it doesn't change our duty toward our spouse and their happiness.  When neither spouse seeks the happiness of the other there is a downward spiral that puts the marriage at serious risk.  If even only one spouse is seeking the happiness of the other, this is less likely to happen, and over time their example may rub off on their partner.

...and in another way, no you are not.
The answer is also 'No' in the sense that each person has the final responsibility for themselves.  Your obligation is to do you part to enable their happiness, after that it is up to them.  If there are mental health issues like depression, you may have to accept (at least for a time) that their unhappiness is not related to their circumstances in life and beyond your power to affect, other than to try and help them find an effective treatment plan.

As much as I wish I could remove every discomfort of any kind that my wife feels, it is not within my power to undo the Fall and make her world a Paradise.  I do what I can, and how she responds to the challenges she faces is her choice.  Likewise I am responsible for my happiness, and she is responsible for doing what she can to facilitate it.

If however the husband and wife are selfish, put seeking their interests first, treat their spouse with contempt, ridicule, verbal, emotional, or physical abuse, take them for granted, withhold help, comfort, support, encouragement, intimacy, or otherwise mistreat them or hedge up the way before them, then they make themselves responsible for the outcome of those actions and the damage they do to the marriage.

Nobody should be a hostage to their spouse's emotions, but they should be a help meet and partner.  And seeking the happiness of your spouse is very likely to result in a greater happiness for you than if you sought your own happiness above theirs.