Conceptually, the working age population is divided into participating and non-participating. The participatings are further divided into working and unemployed (and working-but-looking-for-more-work is a useful category as well).

It is indeed unsurprising that participation drops in a period of high unemployment, but it's not just a matter of mathematics. Instead, it's the effect of people simply giving up looking for work.

That said, it's important to watch out for what the participation rate is measured against. Obviously, the participation rate of the population as a whole can drop for entirely benign reasons (higher percentage of retirees).

No, unemployed people count as participating. Non participating people don't count towards the official unemployment rate. For participation to drop as unemployment grows is the worst possible combination of trends.