Re my background - I've worked both as a self-employed consultant where it was me, myself & I, and I've worked for my current employer as a consultant and now manage a few consultants.
1. Usually not for the time to and from the designated site, but I would bill travel cost (plane tickets, mileage etc, potentially hotels, parking and restaurants). That said, if the client has multiple sites, I'm supposed to be at site A and in the middle of the day they send me to site B, well, they're paying for that time as well as the travel expenses incurred - once the taximeter starts running, it's not going to get turned off until the end of the day.
2. Actual expenses, if you're working on a fixed per diem that works for, say, Chicago and you're working in Seattle, you're paying for stuff out of pocket that you probably shouldn't.
3. That's something you really want to talk through with an accountant, I don't think it's worth becoming an expert in that. Just make sure to keep all receipts, but you'll need that for your client invoices already.
One thing to consider - sometimes clients want you to adhere to their travel policy. If that's the case, you definitely want to make sure you get a copy of it first and it's spelled out in the contract exactly what version you're adhering to. Nothing like finding out that the client's travel policy requires that you only book red eye flights on Spirit and stay in Motel 6 or cheaper, ideally sharing the room with a competitor's consultant. The way I look at this is that I don't want to get rich off the expenses, but the reasonable expenses that I wouldn't have if I was working with someone remotely are covered.
A lot of travel policies are reasonable, but there's always that one client that wants you to stay in a motel so good the roaches just moved out on a per diem that's not enough to pay for breakfast.
Oh, and when it comes to rates (yes, I know you didn't ask about that) - they're paying for your expertise and the time it takes to acquire that, so the hourly/daily rate needs to reflect that. They're not paying for an hour of your time, they're paying for the 20/30/40 years of experience that is leading up to the hour of your time.
Also, if they want a daily rate,explain to them how many hours that buys them. I had some clients in the UK that had the idea they were paying me a daily rate and that would cover as many hours in the day that they wanted. I had to explain to them that I draw the line at 10 hours and then I'm starting to charge extra.