Locals (or experts) I need your advice

Indeed it is possible to find a job while on holiday here as a foreigner... However, you should realize it is far from likely. It's not like jobs in your field are posted on public job forums. Even when they are, it would most likely be in Korean, and often would effectively be closed, as many Korean companies prefer to hire from within their personal contacts.

Speaking of which, your greatest chance would be to use your own contacts in your industry and attempt to meet and greet with those working in your field in Korea. Your personal and professional contacts would be your best chance unless you wanted to go for the teaching thing. This could work as some Korean universities are attempting to go full English with some of their programs. It might be possible for you to find a teaching job from one of these universities.

Well, It is not easy to find a job with just bachelors in engineering here in Korea until and unless you have minimum of 5 yrs of experience. Sometimes, companies like LG and Samsung have recruitment drive for fresh engg grads from civil,mech,chem and electrical into their new offices in europe where they prefer foreign students graduated from the local korean universities. You can keep checking with those recruitment notifications. Besides this what you can do is to just dump your resume in places like 'seoul global center'. Again, jobs for freshers in these places also 1 in 10 jobs. There are head hunters interested in foreign engineers but again they are looking for highly qualified and experienced professionals. Teaching at schools for science and maths also only open for native speakers. These days rules are getting stricter and stronger. Finally salary for fresh grad in mech industry is not more than 2.2k usd.