Can dental sealants cause problems like tooth sensitivity?

Should not. Dental sealants are a thin, preventive coating placed over deep grooves in teeth to prevent cavities. You should have no pain or sensitivity with sealants. If a sealant is too thick, it may cause you to put too much pressure on the tooth when you bite down on it. Over time, this can make tooth sensitive. A minor adjustment to thin sealant and reduce the pressure, should stop any sensitivity.
No. Sealants fill the small grooves on the chewing surfaces of back teeth, preventing bacterial invasion. Cause of sensitivity needs to be investigated and treated. Please see your Dentist.
No, but not panacea. Sealants were one of the most exciting developments in the 1980's. Over the past 20+ years we've found that sealants did not work as well as we imagined. Sealants can still be a great service, but they have to be done immaculately otherwise it's a waste of time. They often need some minimal preparation and thorough cleaning of the grooves before the sealant is applied.
Not usually. Typically dental sealants will not cause an increase in sensitivity to hot or cold.
No. Sealants can actually help reduce sensitivity. A sealant fills in the tiny grooves in the chewing surfaces of the molars thus preventing bacteria from entering those grooves. In placing the sealant an acid gel is used to prepare the tooth surface to bond to the sealant. If this acid gets on very sensitive dentin, it could cause some temporary sensitivity. This is rare.