Of course, you know I"m not a a teacher, but I'm happy to respond.
“Ten dollars award is offered for information of any person injuring this property by order of the owner.”
When we have a "mass" of something (money, time, distance) we think of it as a unit and it takes the singular verb. (I'd write that one as "A ten-dollar award is..."). The owner of this property will give a $10 award to anyone who lets him (or her) know about any other person who causes damage to it.
“The class were distinguished for ability.” I'd be more likely to use the singular, but if you think of the class as many individuals, the plural is okay.
“The regiment was in camp.” The entire unit, the whole regiment, is one thing.
“He, and not you, is wrong.” Take out the part set off by commas: He is wrong.
“Whether he or I am to be blamed?” With an "or" subject, make the verb agree with what's closest.
“Snow and rain are disagreeable.” With an "and" subject, the verb is plural.
“The man or the woman is to blame?” See above with "or."
“This monument was errected to the memory of John Jones, who was shot by his affectionate brother.” -- What's your question here? Apparently John's brother shot him. WITH a comma after "shot" it simply says that John was shot (we don't know by whom) and that his brother is the one who put up the monument. A more likely reading, since affectionate brothers rarely shoot each other.
Would you tell me whether “to blame someone for something” = “to blame something on someone”? -- Pretty much, yes.