Can my employer require me to come into work even if the weather is making it hard for me to get there?
Yes. Your employer can indeed require you to come to work despite severe weather, although a reasonable employer will make allowances for employees who cannot reasonably make it in.
If my company says we should use our own judgment about whether to come in during snow, does it look bad if I stay home?
Reasonable employers don't expect people to put themselves in harm's way to get to work during serious storms (assuming that your job isn't to provide life-saving services). If you judge a weather situation to be seriously dangerous and/or if authorities are telling people to stay inside and off the roads, you should stay inside and off the roads. Those warnings are issued for a reason.
Beyond that, when the situation isn't bad enough to be a safety issue but you'd prefer not to try to commute in the storm, it will come down to the culture of your workplace, what your manager is like and even factors like how often you've missed work recently. If you're unsure, try asking your manager directly: "Is it really OK to stay home during the storm if we're concerned about the weather?" Then pay attention to how she answers, not just what she says. You're listening for the difference between "Yes, of course stay home if you need to!" and "Well, if you really don't feel safe coming in, I'm not going to ask you to, but it looks like we should be OK, so please try…"
If my employer shuts down the office for a snow day, do they still have to pay me?
It depends on which pay classification you fall into: exempt or nonexempt. These are categories set by the federal government. If you're nonexempt, your employer must pay you overtime at time and a half your regular pay rate for all hours you work above 40 in any given week, but your employer can also dock your pay when you don't work at least 40 hours. If you're exempt, your employer isn't required to pay you overtime but also cannot dock your pay, except in limited circumstances.
If you're a nonexempt employee and your office closes because of the weather, your employer is not required to pay you for the days you didn't work. Some employers will pay you anyway, but the law doesn't require it; it just depends on your employer's policy.
If you're an exempt employee and your office closes because of the weather, you have to be paid your full salary for the week, as long as you worked any portion of that week.
If my employer shuts down for a snow day, can they require that I use a vacation day for the time?
Yes. It might seem unfair, especially if you would have gone into work if your office had been open, but your employer can indeed charge you a vacation day if they close for snow.
If I'm on scheduled leave when my company shut down for snow, do I still have to use up a vacation day for that time, even though my company was closed?
This is up to your employer's internal policy. Some companies won't require you to use the vacation time, but others will. If you're unsure, the best thing to do is to simply ask your manager. Say something like, "I'm uncertain how my planned leave works with the snow day. Should I still count the day we closed as a vacation day?"
Alison Green writes the popular Ask a Manager blog, where she dispenses advice on career, job search, and management issues. She's also the co-author of Managing to Change the World: The Nonprofit Manager's Guide to Getting Results, and former chief of staff of a successful nonprofit organization, where she oversaw day-to-day staff management, hiring, firing, and employee development.