No matter what, I want those around me to be happy. I will alter my behavior at times just to make it so. I am never doing this out of malice, but sometimes I think it can be manipulative. Is it wrong to do something just to make someone happy? Especially if other people's happiness makes me so happy?

You have to look out for yourself as well you know.
It was always about them, though, no matter how much they pretended it wasn't. My friend had an insatiable need to be appreciated by others and to have that altruistic view of their self reinforced. Basically, they used their help of others as a source of narcissistic supply, which could be used then discarded when a new source was found.
I'm not saying you are the same as my friend, not at all. I'm just pointing out their strange behaviour as a cautionary example of taking the need to help too far, or even the different subconscious motivations a person could have for wanting to help others.
You asked,
"Is it wrong to do something just to make someone happy?
Especially if other people's happiness makes me so happy?"
It's not wrong and it's normal it makes you feel happy, but in moderation.
If you believe you are being "manipulative" in order to help others, that's not a good sign. When people figure out you've manipulated them, no matter what your original intentions were, it makes you seem as though you cannot be trusted because nobody knows who the 'real' you is. It can make you seem really fake. It also implies you think you are superior; that you are more clever and therefore able to manipulate them.
I'm sure you're fine though, just don't let it get out of hand. :)