From this manga panel:
I am trying to understand「せんで」.
My guess is it's negated する. In Kansai-ben, I've read that しない＝せん, but why is で added (instead of て like しなくて)? Is this how to て-form other ん ending verbs such as せえへん?
It is the negative て form of the verb する. This is common in many dialects and not just Kansai. It is the standard way of speaking on Kyushu. Note that the conjugation in question can be used for two different conjugations in standard Japanese, both of which are applicable but with different nuance:
せんで = しなくて
せんで = しないで
The former means that you do not have to do it, whereas the latter means that you want to do it but do not have to.
Here is a very nice table summarizing a lot of the main dialectal differences in Japanese, where you can also find せん: