Is there a relationship between religion and depression or, alternatively, a link between religion and happiness?

This is a very interesting question, because religion in its broadest definition, comes from two old Latin words which mean to commit oneself to a set of values. In other words, to commit yourself to a meaning in life. In that sense, every person needs a religion. There was a famous man called Victor Frankof , who was a German Jew, and who was arrested in the time of Nazi's and put in the concentration camp. He was also a psychiatrist. He noticed a very interesting thing in the camp. The people who came into the camp and had a meaning in life had some of kind of passion to live for a religion survived physically even better than those who did not. In other words, talking about a religion in this sense, it is really the foundation of a healthy personality.

However, if we take it in the narrow sense, that we either belong to the Catholic religion, or the Islam religion, or the Protestant, or whatever, and ask is there any general relationship between religion in the popular sense and depression, I'd say there is not. It depends on the person concerned and how satisfying that particular growth is to meet the needs of the person for some kind of meaning in life. So, in that narrower sense it depends really on the person who is involved in the religion and what the religion has to offer the person.

The short answer is, in general, no.

This is something which health professionals have considered - there are 1527 papers in PubMed (http://www.ncbi.nlm.nih.gov/pubmed) retrieved by the search string “religion AND depression” - there is no consensus either way

Yes, there's definitely a relationship, but it is an inverse relationship. Religion is a protective factor and there have been hundreds of studies that have found religion is a protective factor against depression and a helpful factor in recovery from depression. As well as the religious faith helping, belonging to a religious community can help in recovering from depression. Social support from a religious community is very helpful in recovery. There's definitely a relationship, but it's an inverse one.
To give you an example, Cognitive Behavioural Therapy (CBT) is world recognised as one of the best treatments for depression. In CBT you learn to challenge the negative automatic thoughts that feed depression. If you have a belief in a loving God, that helps counter those negative thoughts. In depression people tend to have a negative view of the world, the future, and themselves. A belief in a loving God, and a personal relationship with a loving God, helps your view of the world, knowing that God is in control. Even though it might be a struggle at the moment, God is in control. It helps in terms of your view of yourself, that you are a child of God, of infinite value to God and God loves you. That is a huge help if you don't feel very good about yourself, as many people who are depressed do.
In terms of your view of the future a belief that God has a purpose and plan for your life and that your life is guided by God is a huge help in countering the tendency in depression to see the future as bleak. Faith is a big help in recovery from depression. One way of seeing this is that you're in the valley at the moment but the Good Shepherd is going to lead you out of the valley, like Psalm 23. There's been heaps of research, and I'll just quote one, Dr Rebecca L Propst in 1992 did a random controlled trial in which she found that people who used their faith in Spiritually Modified CBT, did better in recovery and had a faster recovery using their belief in God as part of recovery.
I need to qualify this by saying, it also depends how you see God. If you see God as a loving Father, that is tremendously helpful in recovery. If your view of God is someone who's punitive, negative, and angry at you that might not be a help in recovery because some people suffer with a lot of religious guilt, and that can be a negative factor in depression. Sometimes, in depression people find it hard to see God. That's something that can be addressed in therapy. Spiritually modified CBT where you're using existing faith as a strength in conjunction with CBT is an evidence based effective treatment for depression.

Most religions are based on similar spiritual principles, and if used effectively, can help reduce depression.
Depression is often, but not always, caused by negative thinking and feelings such as, anger, shame, fear, despair, hopelessness and apathy. The following spiritual principles can be used to reduce these feelings:

1. Shame- use forgiveness. That is, split the person from your behaviour. Leave, God, karma or whatever else, to judge the behaviour, and forgive yourself becuase your are human and make mistakes.
2. Hopelessness- use hope , stop focusing on goals if you can't see a pathway to the goal, otherwise you will get anxiety, and continual failure can cause depression. Hope, is having a goal, but trusting that you will find a pathway. The pathway comes from your intuition. Worry and anxiety block intuition being activitated. A goal will cause you to compare, whereas hope will provide peace.
3. Anger -use assertivness . Split the person from the behaviour . That is, attack the behaviour and leave the person alone.  Don;t make it personal or take it personally.
4. Fear- use suffering.  The purpose of suffering is to grow when in adversity and you have no influence over the outcome. That is, with fear grow courage, with anger grow calmness, with apathy grow hope. Don't use suffering to blame others, yourself or circumstances. Don't use avoidance.
5. Despair-use peace when you have no influence over a situation. That is, accept it with no responsibility to try and fix it.

Many people feel religion and in particular faith contribute to happiness. Research suggests this to be true.
A quote I like is “Prayer begins at the end of emptiness.” Rabbi Abraham Heschel

Central to any set of religious beliefs are values. One way of looking at Religion is a codifying or enshrining of values (eg: the Ten Commandments).
But as humans we often achieve less than we hope for, behave other than we would wish, or repond differently from our ideal.
When there is an on-going gap between what we aspire to (or believe we should do or say) and how we actually behave or feel, this gap can result in conflict, distress or perhaps, eventually, depression.
So whenever there is unresolved inner conflict there is the potential for distress or even depression.

In the interests of transparency, please note that I write as an athiest who accepts the sincerely held beliefs of others

Religion as in the latin meaning of “religio” or “connection with the Divine” is a road that definitely will lead to personal happiness by adding a spiritual dimension to the individual that helps in creating meaning and purpose to h/her life. Men and women who have this connection/faith are less likely to fall into depression and if they do, will come out of it more easily and rapidly.
Religion as a creed or a dogma can have the same affect as long as the believer accepts without questioning too closely the rules that bind it.
On the other hand, adhering without questioning a specific dogma can lead to strong repression of feelings such as anger, grief, sadness etc…and therefore to depression.
Hope this helps.

Depression in relation to religion, could result from abuse from religious people and institutions.

Depression could result from religious misguidance, where there is an awareness of guilt, without the awareness of forgiveness and love occurs.

The link between religion and happiness is where  there is forgiveness and love, these are keys to happiness.