For the second, generally make sure the documents you are using for entry are valid at time of entry; in particular a lot of countries require 6 months validity on passports.
I know multiple people who reported their expiring passport as stolen, just to be able to keep it (usually for purposes of having proof that they've held certain visas in the past).
Eh, you want to watch out doing that - some countries get real suspect if you have ever reported a passport stolen in the past.
My girlfriend needed her old passport for travelling with an Interrail train ticket because the passport expired between the time of ordering the tickets (and submitting the document number) and travelling.
It's definitely a thing across various ex-Soviet countries, and likely others as well.
I've heard of people's visas getting denied because their passport would expire before the end of the visa.
This will soon be the case for me - can you explain how that works? Do I just bring my old and new passports to the border and show them both?
>A US B1/B2 Visitor Visa will generally get you into an additional dozen or so countries
It can work the other way too. Due to a mix up with dates I had to get one cancelled (and a new one issued) so the original in my passport has a huge red "Cancelled without prejudice" stamp across it. The border guard at Honduras was not happy - he figured if I wasn't allowed into the USA he didn't want me in his country. It took many, many hours to explain I had another perfectly valid USA visa.