no , it’s means you can come and go as much as you want , just saves you applying for a re entry every time you leave .. if you intend too or maybe leave the LOS on many occasions, then it’s worth getting a multi entry
No .. ( well even if they did you’d have to go back to the airport the next day anyway to collect it , longer if you did it on a Friday ) but I’ve never heard of airport immigration doing anything like that .
imo , when you weigh up the cost of travelling back and forth effectively 4 trips , I don’t know where you are , but for me personally cost effective via an agent .. it took 10 min .. and they called me on its return, asked if I wanted it deliver to my address or come into office to collect , which was the option I chose .. 😉👍🏻