No- pensions take money from you while you work, put it into savings (investments) and then use that money to pay you when you retire. A properly ran pension fund is not dependent on current workers to pay current retirees, and if there were no current workers, there would still be money to pay retirees.

Unfortunately, a significant number of pensions have large amounts of unfunded liabilities, which means they may end up bankrupt or convert into pay-as-you-go.

Most pensions at least pretend to be equivalent to a retirement savings account. Defined contribution pensions are effectively a savings account, and defined benefit pensions pretend to be similar (but are usually not).

Defined benefit pensions are usually used for government roles and some large unions, where the unions can get large increases in total employee benefits without having the true cost show up on the organization's books. The union and employer generally agree to use unrealistic numbers for returns on investment (>=8%/yr).

Yes, although I think the initial lag for pensions to start paying out is longer. First SS retiree payed in for only three years. https://www.ssa.gov/history/imf.html