What is (or are) the most common and natural way of saying "to pee" and "to poop". And a follow up question might be if there are any connotations in realtionship to keigo.
In this context, what sounds natural to one will sound awkward to another. Still, some expressions are used much more frequently than others.
To refer to urination, おしっこをする, 小便をする, ションベン（を）する (mainly used by male people) are probably the most common 'natural' ones. To refer to defecation, うんこ（を）する, 大便をする are relatively typical. There are tons of other ways to refer to these in an informal context, though.
I've never heard anyone explicitly refer to the act of urination or defecation within a context where polite language is appropriate. This person suggested using お小水 for the former and お便 for the latter. These refer to urine and feces, respectively, though, and would be used in a roundabout way to refer to the act of producing them.
Usually people will try to avoid specifying what they or other people actually use the toilet for, especially in a formal situation, and the only exception that springs in mind is in a medical context. A doctor might use informal language to a patient, but when discussing it in a more scientific (and thus formal) light, they would tend to use 排尿 and 排便. You could make these compatible with keigo as ご排尿 and ご排便, e.g. 「ご排便中、誠に恐縮ですが」. It sounds a bit funny though.