Difficult to say except your company will give you the required documents for the visa and you may likely need to go to the RTE of a neighbouring country as the required documents are fewer and less arduous than submitting in LOS.
Bear in mind, your WP will likely need to be sorted subsequently.
Yes, you must stamp in and out of another country. If you have a history of border bounces, they’ll question you at some point and deny you entry back to LOS at another…. You’re at the discretion of the officer.