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I know that the characters can join other factions as a HQ choice, but do they stay as their own faction (i.e. Cawl is mechanicus) and thus are unable to start the game in your factions transports? Is this stated anywhere I am not sure.

They stay their own faction, because you're not told otherwise.

Out of curiosity and a need to prove my friend wrong, may I ask where it tells you what faction they are? Is it in their rules or the fall of cadia book?

A unit's Faction is shown by the symbol at the top middle of that unit's dataslate - for more information on Faction symbols, check out page 118 of the rulebook.

Depending who you ask, you can put him in transports, however due to GW slowly eroding/evolving their allies rules and how they interact, you can take some detachments to give hima transport. Such as the Casteelion detachment, or using the Inquistor agents detachments, as that book has a whole bunch of "cheat/breaking of standard detachment rules".

It looks (and I hope) that GW is going to have the detachments become the new ally system, instead of faction, but only time will tell.

You cannot give him a dedicated transport, since the Cult Mechanicus faction does not have any transports or dedicated transports.

You can of course have him embark into an allied transport during the game, but he cannot hitch a ride in a droppod or start the game inside an allied transport, as clearly stated in the now final FAQ.