Is it inappropriate to email a senior academic with whom I have no pre-existing relationship, asking to check/inform my work? Is it inappropriate to mail a senior academic (e.g. professor), whom I don’t know, asking –


to check  Lemma, Theorem ( less than 3 to 4, short/ little)
how a graph can be constructed (with given conditions), this might require his ‘intellectual time’ as the class of graph is unlikely to be familiar. 
reference related 1,2 .


In general, if I need 1,2,3 as described above how I communicate with an academic over e-mail?
Inappropriate or not, you aren't likely to get a response. Professors tend to receive a lot of email. If they don't have a compelling reason to respond, they might not.

You're asking them to put time and thought into a response to a stranger. The best course would be to talk in person to a professor whose class you've taken. Ask if they would mind looking over your work, and get feedback in person.