Serotonin is not the most important transmittor in stress. However, bananas contain serotonin and have a calming effect which mothers with babies make use of.
Probably nothing significant. When we're talking about mental health, neurotransmitter such as serotonin works in the junction between one neuron axon and another neuron dendrite (it's called synapse). Basically, it's a substance that is used to help "the cables" to pass the signals along it's pathway. When we are talking about "increase" or "decrease" of serotonin that can affect mental health, we are usually talking about serotonin in these trillions of small gaps. The release and reuptake of serotonine is regulated by neuronal cells, but it can be manipulated by drugs.
That's why supplying serotonin from external source might not mean too much. And serotonin is even blocked by Blood Brain Barrier that it cannot enter the brain from bloodstream. What we can alter is the release of serotonin to synapse, decrease serotonin breakdown inside neurons, and stop the cell from retaking the serotonin that has been released in the synapse.
Do you mean that the importance of Serotonin is assisting and cooperation of other nerve signals or other transporters, but not because its action itself ?
No. Serotonin does an action on it's own, but the action is done in a very small junction between nerve cell. The question is how you'd like to measure the amount in such a small gap during a brief activation.
How do you plan to increase it? The activity in the small junction might not reflect the total concentration