Thank you for refering me to SQL::Abstract, this is where the WHERE clauses are documented. Furthermore, the suggested NOT IN works well, great!
Regarding the two cases listed in my question, I found that

1. The proper code is $schema->resultset('my_table')->search({ id => {-in => [12,13,17]} }). Note that the r.h.s. of -in is an arrayref, not a string as I guessed initially.
2. NOT is not yet possible. On SQL::Abstract, it is explained that there is not (yet) any explicit support for SQL compound logic statements like "AND NOT".