Your loop looks correct apart from using NR (number of the current record/line) in place of NF (number of fields in the current record/line), but it would be easier to count with gsub():
$ awk '{ for (i=2; i<=NF; ++i) sum+=gsub("e","e",$i) } END { print sum }' file 6
The gsub() function returns the number of times it makes a substitution.
In Perl, you would use the tr operator in a similar way
$ perl -ane 'shift @F; map($sum += tr/e/e/, @F); END { print $sum, "\n" }' file 6
Or, you could just use the other basic utilities from the tool chest:
$ cut -d ' ' -f 2- file | tr -dc 'e' | wc -m 6
This cuts off everything before the first space character, deletes everything that isn't an e, and then counts the number of characters that are left.