Is the reason its "foot" instead of "feet" because of the verb? So if it had been feet it would then be: Peter के पैर में दर्द हो रहे हैं?
For feet, you would use the plural form 'पैरों में'. The sentence would then be पीटर के पैरों में दर्द हो रहा है। It is still 'हो रहा है' and not 'हो रहे हैं' because the verb is referring to the noun 'दर्द' and not the 'पैर'.
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'.
To add my guess, Hindi makes it into four words:
Rain make -ing is.
Or, in the original sentence:
Peter of foot at pain make -ing is.
Am I right that the हो here is not the same हो as the one that usually comes with "tum" ?
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).