What properties such as shape (rectangular rather than the normal
circular), rotation speed, material, presence of a paper tag and more
may cause a CD to explode in a drive. Please provide a sense of
probability distributions of explosions vs. a measure of these
parameters. More precisely, is there a threshold to respect to avoid
incidents with a 99% probability.

I did not find enough information for a complete answer, but I will
post some in case it's useful to the questioner or another researcher
working on a more complete answer.

I think the term "shatter" is more likely to be used than "explode,"
so I searched Google for

"compact disc" OR CD shatter OR shattered OR shatters OR shattering

The primary factors contributing to a CD shattering are rotation speed
and cracks in the disc, according to most sources.  48x is usually
cited as the speed where the risk increases, though there are reports
of discs shattering at lower speeds.

A marketing manager at drive manufacturer Lite-On says the risk of
shattering at 48-52x is only 0.01 to 0.02 per cent.

http://www.pcworld.com/news/article/0,aid,105458,00.asp

Other factors mentioned with less frequency:

--age of the disc.  The computer company Gateway says discs get less
flexible with age.

http://support.gateway.com/s/CDROM/shared/shatter.shtml

--disc out of balance due to low-quality manufacturing

--disc out of balance due to paper label

Other links:

http://www.rm.com/safety/Default.asp

http://www.cdrlabs.com/phpBB/viewtopic.php?t=7805

http://www.theage.com.au/articles/2002/12/09/1039379779472.html

--efn