May I ride the brevets out of order, a 400k brevet before a 200k or 300k brevet, for example?
In general, yes, but the rules imply that the brevets be done in order. This is a matter of tradition, training and logical progression, as much as anything. However, note that beginning in 2007 the ACP no longer insists that the four qualifying brevets for PBP be done in order. Other 1200k events will have differing requirements--check with the event director if you have an unusual sequence of brevets planned.
If I miss a brevet can I make it up and still enter PBP?
Yes, you must simply do another sanctioned brevet of the same distance elsewhere.
In cases of extreme duress you may do a brevet of the next higher distance to substitute for a missed brevet, e.g., if you missed your 200K, then you can do two 300K brevets instead. You must work this out with your RBA before you undertake the replacement brevets(s).
If I abandon a brevet, can I make it up?
Yes, but under the same guidelines outlined in #3 above.
If I experience a mechanical mishap which prevents me from continuing the brevet with my original bike, may I use another bike to finish the brevet? Or do I have to repeat the entire brevet?
The rules do not preclude using a replacement bike in case of emergency. You can buy or borrow another machine anywhere along the route, and you can get roadside help from your fellow randonneurs entered in the event. But note that a replacement bike, or repairs, can only be given by your personal support crew at an official brevet checkpoint. Anyone caught getting personal support in between checkpoints from a their support crew is subject to time penalties or disqualification. Depending on the circumstances, it might be a good idea to somehow get yourself back to the previous checkpoint you last went through. Try to get some help there-you might find another rider who DNFed and you could borrow his/her machine, or the parts and tools you need. Going forward isn't such a good idea, unless you get an event official to observe that you resumed cycling where your first bike failed-not always such an easy thing to prove. Obviously, starting the brevet on a thoroughly reliable bike is an important consideration. Avoid using super-lightweight bike parts or frames that may fail prematurely, or worn-out parts that should have been replaced long ago.