At speed, if they have to. They'll move to the outside of the pack, use the stretchiness of their riding gear to get their parts out, get out of the saddle, turn sideways and let it flow.
Then they'll hope that their relief will enable them to catch up with the pack again.
on the side of the road.
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They don't, unless they absolutely have to (in which case, they have to stop and catch up. Once upon a time, when professional cycling was a more gentlemanly sport, the entire race would stop and wait when a rider needed a toilet break but those days are long gone - see Alberto Contador's controversial decision to keep going when Andy Schleck's chain came off in the 2010 Tour). To urinate, they just drop back from the front a bit (they're still gentlemanly enough not to want to urinate on other riders), roll up one leg of their shorts and do what they need to do. Watch carefully this year if you see a rider fall back from the front for no obvious reason and you might see a little bit more than you want to.
They either get off to the side of the road. This is very rarely filmed, just as a convention. In the heat of the race they let go on the bike.
Now Jan Ullrich famously....
No. You have to be a part of a team to be allowed to race in the Tour de France
In the 1919 Tour de France, just 11 Riders out of 69 entrants finished. This was also the slowest ever tour. http://www.bikeraceinfo.com/tdf/tdfstats.html
Between 18-22 teams with 9 riders in each.
Usually by team buses. Longer transfers, they fly.
Yes