I carry a blowout bag even in a very short sprint. No sense getting a DNF for something so trivial. Yes, my bike time will be shot, but I can still try to get my best run time, etc.

Depends on how much the race is taking out of you and your goals. Personally, I like to be able to change it, accept the misfortune, and keep going. But the guys out there for the $$ can be smart to say "It's not my day" and to drop out and minimize the time they need to recover, or to sub in a workout that will help them more -- although that is more true of longer stuff, it does play a small role at any distance.

Esp. if you borrowed the bike or wheels :p
I think my wheelguy would be less happy to hook me up with with wheels again if I ever tried that!

Always finish the race. Plus, in a sprint - even if you lose 10 minutes fixing it - there will always be people slower ;)

But, more importantly - it teaches you how to fix the issue under pressure, so for larger events you won't be as frazzled by it.

i was doing an off-road duathlon recently when my handlebars on my bike became very loose. i stopped to tighten then but within moments they were twisting again so i tried to ride anyways, which caused me to fall and mess up my front derailler which kept me from pedalling....so i walked off.
at the end, and to this day i wish i had coasted the downhills and run it up the hills.
do what you can to earn that race number. if you have to fix your tire twice and come in last place, you'll be happy you came in.

I dont carry a spare for sprints. Oly and longer, yes. I havent flatted in a sprint yet. Might change my tune after I do. I did flat during an Oly once, but didnt realize it until I was done.

I am pretty sure that if you don't want to DNF, you need to cross the line into T2 with your bike, either carrying it or pushing it.

Ya, you must complete the bike leg with your bike. Carrying it is fine.
A guy at an IM was hit on the bike - and carried/ran with it for 6 miles to finish the bike leg.

As others have said, it has to be with you. How isn't a concern as long as it's under your own power.

It's similar to the swim, and the fact that if it's shallow enough you can legally walk through the water, as I've been told.

OK then, what is the best (smallest/most compact) set of tools for carrying with you during a race?

Really now that I think about it, I guess I have been fortunate enough to have not had a flat away from my house yet since I don't carry a kit anywhere. Everytime I have had a flat its always been getting ready to head out and notice that one of my tires is flat and one time they were both flat. Of course now I have just jinxed myself.

When I train my saddle bag has: 3 CO2s (12g unthreaded because they are cheaper -- at wallmart you can get the ones used for paintball guns), a tube, a stick-on patch kit, a CO2 pump that works with unthreaded CO2, 2 tire levers, and a couple dollar bills (to boot the tire if I get a sidewall issue).

I just don't want to get stuck or have to pay for threaded CO2s for each flat, and I don't mind having to ride home without optimal tire pressure.

When I race, I carry: 2 CO2s (16g threaded), a tube, a stick-on patch kit (sometimes), a Microflate, 2 tire levers, a couple dollar bills, and a Vittoria Pit Stop. I just want to be able to fix it myself if possible, get the best tire pressure I can, and keep on racing.

My training saddle bag is huge. Mini pump, 2 spares, tire levers, Park 8 mini tool, room for gatorade powder, extra food, and a really compressed jacket for the winter.

My racing bag- old GT bag that fits in the palm of my hand. It can hold 2 Co2s, one spare, patch kit, and two levers.

Not to get too off topic, but I tend to keep the emergency/drink stuff separate -- I use a ziplock bag (snack size) in my jersey pocket that has an older drivers license, a medical insurance card, a list of phone numbers, a credit card, some cash, and my cell phone. When I did need the emergency info (before I started that system) my bike wasn't with me (ambulance = someone took my bike home and gave a phone number to get it back) and I wasn't in a frame of mind to have grabbed anything off it before hand. I also have a road ID on my bike computer, but again, that would have still be less useful at the crash. One of the worst parts of the crash (well, mentally) was that I couldn't remember phone numbers, someone had taken my cell to try and get ahold of family after no one was allowed to go with me, and I was in some other county in a hospital -- I felt soooo alone (when I could think again).

I keep the ziplock with all the contents but the phone in my helmet :)

Look at the rules again I found an even more explicit rule for this one in the bicycling conduct section:
5.2 Forward Progress. Participants shall not make any forward progress unaccompanied by their bicycle. If a bicycle is rendered inoperable, a participant may proceed on the cycling course, running or walking, while pushing or carrying the bicycle, so long as the bicycle is pushed or carried in a manner not to obstruct or impede the progress of other participants. Any violation of this section shall result in disqualification.

I have a Hurricane Quick Shot Pro (or something similar enough that from that image and description I think that is it).

Oh, and someone earlier mentioned putting a CO2 upside down in teh pump to save space in the bike bag, and one thing I like about my CO2 pump is that the bottom actually has a hole that the neck of the CO2 fits out -- so I can screw the top on the pump completely while the CO2 is upside down. I find that helps reduce rattle. Another rattle-reducer (while I'm already off-topic again ;)) is to wrap the CO2 that goes upside down in the pump with the 2 dollar bills I keep for booting tires :)

I've heard that the CO2 is a one-shot system, is that the case, or can you adjust how much goes in? Meaning, can I stop and then restart again?

Just sitting in the helmet... I mean at home, as a reply to a comment about forgetting to put stuff you want in your jersey pocket before leaving home. That's how I remember it.

I did a race over the summer and had my first flat on the bike...it was a 10 mile ride..5 out 5 back...and i was crusen on my aero bars becuase i had a bet going for the bike ride....i was either looking too far down the road or lookn at my speed when i suddenly saw a 2 inch deep 1 food diameter hold in front of me...i was able to get my arms over to the sides of my bars but couldnt swerve away from it, so i lifted my front wheel off and let the back wheel hit.....5 seconds later i hear...pshhhew pshhhew pshhew.....i yelled a few foul words and then ran the last 4 miles of the ride barefoot in because i dont take replacements for sprints.....i then did the 5k run and you would not believe the inspiration that gave people during the run.....they would say thank you for not giving up because they saw me running it in then going by them on the run.

Never DNF on your own unless it's for medical reasons. Make them pull you off the course if you are over the time limit.

"To give anything less than your best is to sacrifice the gift"
-Steve Prefontaine

No matter if you're first or last, you have a gift, you are out there.