Does anyone know how many murders a sheriff can handle in a small town (20,000 or so) before a higher authority steps in? I am thinking something along the lines of a killing spree, three or maybe four murders in a span of a week or so. also would it be the responsibility of the sheriff o notify said higher authority or would it be his/her call.
In reality not many, in fiction as many as you like (see for example John Sandford, Winter Prey or Robert B Parker night passage) In reality a small town sheriff isn't set up for a complicated murder investigation (unlike where Billy Joe shoots Bubba over profits from the meth lab and is still covered in blood when the cops arrive), so just one killing with no obvious suspects might be sufficient to have them calling the state police.
Local authorities generally get to chose whether they want to involve higher level law enforcement agencies (like state police or the FBI). A spree killing would probably put pressure on them to call in aid, but it's still their choice. I should point out that a town of 20,000 or so probably has a decent sized police department, so if the murders take place in city limits, they'll be the ones with jurisdiction. Very small towns (as in very low thousands) may not have a police department at all, and thus rely on the sheriff's office and/or state police for their law enforcement.
Officially chiming in from a small town . . . Less than 1,000 people. We do not have a police department. The county will occasionally send a Sheriff's Deputy out on patrol. There are only about 38,000 people in our whole county; the county seat has about 24,000 people; they have a police department and handle about 1 murder a year. Three murders inside of a week would have them begging for all the help they can get, plus they'd probably lobby the governor to call out the national guard and put the county under martial law until the killer was caught. Just sayin'.
Overall, the US has something around 1.1 million cops for 323 million people = about 1%. I recently checked out a small US town (Harlem, Montana, pop. 808) which has a privately-run sheriff's dept with "between 25-49 employees" (but this probably covers the neighbouring small town of 1,200, and all the farms for a few miles around). Lower cops per head than the national average, and probably most of those are patrol officers on shifts. Maybe a handful of detectives. And probably limited forensic capability. This last is probably why a small-town sheriff would call for the Feds.
