It really depends on what a snake symbolises to you personally and under what cultural context you grew up and live in. As such, it's up to you whether the following makes sense to you or not.

In my mind, in your dream at a young age, the snake wrapping around your legs with open wounds could potentially symbolise a healing process or something you were doing to cope with your situation. Interestingly, the caduceus is a symbol where a rigid staff is coiled by one or two serpents and there are some visual parallels here with your leg and the symbol. If you are not familiar with it, it's still quite possible that you had seen this symbol at a young age in a place such as a pharmacy or hospital, on their logos or staff uniforms and that there were some non-conscious associations going on.

Also, the snake can typically considered to be a male (phallic) symbol, so perhaps your dad was more protective of your well-being in a way that your mom wasn't, since she was the perpetrator of the injuries caused to your leg? Another possibility is that a belt and a snake have some similarities in how they appear, often leathery, coiled or tangled in some way; belts also have a "head with fangs" (the buckle) like a snake does, so under that type of interpretation it can also be that the snake is a representation of what a belt may have felt like to you, like a threat of some sort.

Your more recent dream could have something to do with relationships or how you engage in them. The tattoo the guy had, was it also of a snake?