A person will share roughly 3% of their DNA with a second cousin. A person shares 50% of their DNA with their father. If you have the luxury of testing the autosomal DNA of both the person and the second cousin, then that will obviously give you the answer. However, I expect that is not the case.
If any other close relatives (siblings, parents, aunts/uncles) of the second cousin have tested, this could also give you an answer – you would expect to have a stronger match with these people if your second cousin is also your father. If you have the opportunity to test other family members (on both sides of the family), I would encourage you to do so, as this will help narrow down any genetic matches to one side of the family.
Ethnicity percentages will be a quite unreliable and subjective means of drawing a conclusion, and would only really be helpful if your second cousin/father had a particularly unique genetic history compared to the rest of your family (i.e. ancestry recently from another continent). If not, then I would tend to ignore ethnicity percentages for this purpose.
A child born between two second cousins would not, in most people's estimations, be considered "close inbreeding" (I have a number of first cousin marriages in my family tree). However, there are tool available that may be able to detect such a relationship based on the results of an autosomal DNA test. Each chromosome has two alleles. If a person has significant areas on each allele that are identical (runs of homozygosity) then this would be indicative of one's parents being related. Gedmatch has a tool call Are Your Parents Related? which tests for this.