दर्द is a noun and not a verb. So, you add the verb हो रहा- (happen) to it. So, the Hindi sentence would literally translate to 'Pain is happening in Peter's foot'.
A similar construction can be seen when you translate 'it is raining'. In Hindi, it would be 'बारिश हो रही है' which is literally 'rain is happening'.
I believe you are right. The हो in हो रहा है is the root of the verb होना (to be/exist/happen). The construction is the same as खा रहा है, "is eating", in which खा is the root of खाना. So, in हो रहा है you have "assembled" the continuous form of the verb होना, second (tu)/third person singular, as indicated by है (again a form of होना), while in हो used with "tum" you are using the second person plural of होना (which can also be used as a polite alternative to "tu", when addressing a single person).