To address the "mix and match"---in the classes of racing I have been involved in the rules stipulate the same size all 'round. Not necessarily the same mfgr but even if the sizes were identical, the coumpound variances would coumpound (excuse the pun) any handling problems. Besides, usually we have some contractual agreements with the company whose tires we choose to run. Not disagreeing with you Greg, just adding my "take".
As to "run in"---If I choose to start a race on "stickers" then I can expect the tire to come to operating temp very quickly and give max performance early. I can also expect the temps to continue rising until the tire overheats and grip "goes away"---a one heat cycle tire, as it will not be very good the next cycle. On the other hand If I go out on a new tire and bring it to temp (4 or 5 laps) and let it cool (known as "scrubbing in"

then the same tire will be good for several heat cycles. The number may vary with tire compound and mfgr. but, all will last several sessions with good grip. Most race tires have an "arrow" indicating which way the tire should be mounted but after the first heat cycle it is somewhat common to run them in any position that is advantageous, based on their circumferance and/or wear patterns. Not necessarily condoned by the mfgr, but, none the less, a common practice. ( I have never had a "tread" unpeel, as it were, with bias tires but I don't practice this on MY radials, but some others do.)
The difference in the tires I chose to put on my antique Norton was a combination of need vs availibility. It was necessary to relace the standard 20 inch wheels of 1948(virtually no tires available) with a 19 rear and a 21 front. The tires were a known quantity from previous use in years past combined with the fact that I am now "two days older than dirt" and I ride rather more conservetively. Mixing brands has always been common practice as 'Swall' indicates. Still, I run in the tires as well as brakes just like I have done all my life, just a habit.
Rod