The answer to both questions is, it depends.
Depending on how the plan document is written with regards to payouts, it is legal to have the plan structured so that payouts are only made once annually, with everyone who termed during the previous year receiving payout at that one time. That means that if the payout is, for example on January 15 and you quit January 20, you'd have to wait a year for the payout. I do not believe the plan document can legally require you to wait any longer than a year. However I once took over administration of a plan that was structured that way (it got amended VERY soon after I took over); the plan document had been approved by the Feds and was managed by a VERY reputable company, so I can verify for certain that the year-long delay is legal. It doesn't happen often, but it can and does.
Specifically who manages the plan would determine who wrote the check, although I can't say I've ever heard of the check coming from the employer. While it's a bit unusual, if they're self administered I can't offhand think of any law it violates.