Regarding the clans and races:
As I recall, the bloodname of any Clan mechwarrior is traced through the 'mother' so a trueborn of any race with the proper genetic mother would qualify for any given bloodname.
Freebirth scum, on the other hand....
Ummm.. Well they are traced, yes, and there's no mother for True Born warriors... Unless you consider their artificial womb they are born from as a mother lol. The 800 Blood Named warriors, the originals that is, are where the genetic lines come from, each True Born warrior is born of one of these main lines of genes, a long with multiple other warriors gene's that are used to create an amalgamation of DNA so as to avoid just one genetic line of warriors with no advantage over the previous one, it would basically make one warrior so once a clan figured how to counter one warrior they could kill them all as they all would react the same in each situation, thats why they mix max, even sometimes trade some genetic material to other Clans for certain things to mix up the gene pools.
After the name is traced back it does NOT get given to the soldier, no matters what, not even Khan's of the Clans get given birth names, and there are some Khan's and SaKhan's, Galaxy Commanders etc that have NO blood name! They go through the Trial of Bloodright, there are currently only 760 surviving Blood Names in the clans, only 25 warriors can hold one of these at the same time, 19000 warriors MAY hold blood names in the clans, but there will NEVER be this many due to Reaving.
The Trial of BloodRight is made up of however many warriors the clan chooses to allow, these warriors must have sponsors, certain warriors are assured to compete in the tournaments but for some its not so certain, these 'Mech Warriors must go through the melee, a 30+ giant freakin bash where warriors fight till the last man or woman, this person gets a slot in the Tournament.. As Diana Pryde, a Freeborn warrior, had to do to earth the most prestigious blood name of the Clan Jade Falcon. After this Melee, all the warriors int he tournament fight, often to the death until the final round and then the victor recieves their Blood Name, the highest honor in the clans, higher than being SaKhan, Khan, or even the ilKhan, this is hands down the only drive for a warrior, to achieve this Blood Name and to contribute their genetic material.
Freeborn Warriors also can get Blood Names in all the clans as there are FreeBorn blood names they compete in the same fasion but no one gives a darn about them. But some clan FreeBirth warriors can compete for even the most honored Blood Names, but only if their lines coincide with the genetic material of the line, and sometimes entire Blood Names are simply created for anyone to compete in.