Let me start by saying that over the past few years I've collected over $3000 in airline vouchers, plus a few hotel and meal vouchers, from volunteering for oversold flights.
Unfortunately the specific questions you've asked are hard to answer, as they vary between airlines, and often even individual airports/staff.
In general all airlines will announce that they are looking for volunteers at the gate. Depending on factors such as how many people they are oversold they may start announcing this an hour or so before the flight, or they may only do it closer to the departure time. In addition, some airlines will offer to add you to the volunteer list during check-in, especially when using an airport kiosk to check-in.
There's also nothing to stop you simply going and asking - especially if it's an airline where you can view the seat-map and see that the majority/all of the seats are taken. Although the seat-map isn't necessarily a good indication of how busy the flight it, it can still give a good idea, especially if the flight is full. I've been bumped off several flights where they never announced that they were looking for volunteers - I got in before they had a need to announce it!
As far as compensation, it varies a lot depending on the airline. There are NO regulations on what they must offer for volunteered denied boarding, but most US airlines will start at a few hundred dollars, and then go up from there depending on the length of the flight, the length of the delay (e.g., you should get more for taking a flight 24 hours later than one 2 hours later), and how desperate they are to get people to volunteer.
Sometimes you can ask them to sweeten the offer by putting you in business/first class on the later flight, although doing this may end up with them offering you less compensation.
The compensation they offer will almost always be in the form of vouchers for future travel - not cash! Sometimes there will be limitations to these vouchers (the most common being that you often can't combine multiple sets of vouchers to use on a single flight).
Frequent Flyer status sometimes help - again it depends on the airline and the specific staff. Even when the airline doesn't have a specific policy around status, some gate agents will prefer to "bump" someone with higher status - partially because they see it as a benefit (they are paying you, after all!), but also because bumping a frequent traveler is likely to result in less problems.
Many airline/staff will also give preference to people travelling without checked-in luggage. If you do have luggage checked, then depending on country and airline policies, it's possible that your bags will travel on your original flight, which means they might sit around in the destination airport for some time before you arrive. I once volunteered off a flight to Miami and instead flew to Fort Lauderdale (about 20 miles away), and then had to drive to Miami to pickup my luggage as they hadn't been able to change it to the new flight.
There are some big disadvantages to "volunteering". Frequently you will be told to remain in the gate area during boarding in case they need you, only to be told that they don't need you as another passenger didn't show up. You then end up being the last person to board, which may result in trouble finding overhead space, etc. If you do get moved to a later flight, there's a good chance you'll end up finding yourself in a middle seat given how late you were booked onto that flight. Occasionally you can hit agents who make mistakes when rebooking you onto the new flight (especially as they are normally busy with boarding your existing flight/etc), and find when you get to your new flight that you don't actually have a confirmed booking on it!
Personally over the past few years I've volunteered to be bumped dozens of times, and been successful about 8-10 times, including one occasion where I "double-bumped" as the flight they put me on after the first bump was also overbooked!! On some of those occasions I've had replacement flights that landed a few minutes after my original flight, and on other occasions it's been the next day (with a hotel provided). Sometimes I've ended up at the same airport I was supposed to go to, and sometimes to alternate airports (Miami v's Fort Lauderdale, San Francisco v's San Jose). I've had compensation of everywhere between $200 and $600, and on occasion I've received First class on the replacement flights - although on at least one occasion I've also given up a first class seat and ended up in economy on a later flight!