Can my employer require me to provide a car, at my own expense, to get home from night work-related meetings not accessible to my normal public transit?

Your employer isn't legally obligated to provide you with any transportation at all.  There are many people who use their own vehicles for work-related travel, other than commuting, whose only "compensation" for that is deducting the expense from their income for tax purposes (the IRS allows you to choose between deducting the actual cost, which can be a bit involved, or a flat per-mile rate), for trips between two "work" locations.  You're going to have to figure out whether your job is worth keeping, with this extra expense.

There is no legal standard or rule requiring employer reimbursement of costs like this. (There are rules governing when costs may be taken as business expenses for tax purposes by a company or an individual; however, those rules do not require reimbursement.) Unfortunately, it is up to the company to decide which expenses, and what portion of them, they will reimburse. That means that yes, you must pay the non-reimbursed costs. I suggest you look at the rules governing when you can claim mileage or other travel expenses on your taxes and at least work with your employer to structure the travel in a way that allows you to take tax advantage.