Am I responsible for the happiness of my elderly mother? She is in a SNF and I constantly worry about her emotions.

If you are worried about her emotions, you need to get her to a geriateic psychiatrist.

Your mother's happiness comes from within. If she is depressed, anxious, ruminative, she may need meds to help with that.

I think that you likely understand very well that you aren't responsible for your mother's happiness, and that there is very little you can do to "make her happy". In all truth life isn't about "happy". There is good and there is bad, and for our elders, esp in care of hospital, snf, LTC, it is definitely not a happy thing. There is loss after loss after loss from dignity to bodily function, and there is no upside coming. This downward slide continues until death.
I think what we ask of our elders often enough is to PRETEND it's OK. It isn't. And why should they pretend it is. There is also the fact that people have differing attitudes to life. Was your mother EVER a really happy positive person? If not, then she sure won't be made better by the aging process.
You might consider some therapy to help you understand that your own satisfaction in life, your own choices, your own "happiness" isn't dependent on your Mom.
I sure do wish you good luck. You may feel grief in witnessing your Mom's losses and consequent her own grief, but don't mistake that for the other G word. That is for evil felons, and I doubt you are that.
I sure wish you the best going forward.

Only if you can be responsible for my happiness. And if so I would like a Pint of Ben and Jerry Pish Food delivered by 7:30 PM. And the bathrooms cleaned some time this week, I am not picky about what day.

No one is responsible for anyone's happiness other than the person themselves.
You can be responsible for her well being. For her basic needs clothing, food, safe housing. That is being done by her SNF, and you are an advocate for her in that respect. When you visit you can do what you can but that also is mostly up to her.

The facility can care for her, can try to get her involved as much as she is able. But they also can not "force" someone to be happy, participate.

Of course you're not!!! The only persons happiness you're responsible for is your own. Your mom is a grown woman, and has the choice (like we all do) to feel how she wants to feel. That's on her not you.