Position applied for: Investigative Team. He would not be particularly interested in a leadership position.
Is your character psychologically fit? Please elaborate: Very much so, since this is not all the different from what he does for a living. He spends most of his time in high stress, save the world type of situations, unless he's effected by it personally - say if someone he cares very deeply about is in life threatening danger. (And since at the moment that's only Jack who is immortal he's not likely to have to deal with that at all.) He operates on both the research and operation sides of Torchwood, but is usually more comfortable handling the investigating part of things.
How much of this would the interview team be able to read?: He comes off as a very helpful, proficient butler type, which is ore or less what he is. He's likely to be quite truthful, unless its a very private, personal question he doesn't think relates to the job. Normally he'd be a bit more cagy about the whole alien hunting thing since Torchwood is top secret, but since it can't be much weirder than their current situation he doesn't really see the point in hiding much, so it should be fairly easy to get a good read on him.
What skills/knowledge/powers/experience does your character possess which would be an asset to the position(s) they are applying for? Do you need papers alphabetized, coffee made or someone to keep a schedule organized? Because he excels at that sort of thing. Frankly he's just a very good butler/secretary/assistant. He also usually handles the research side of things, is pretty proficient with a wide range of alien technologies, and can handle a gun. He has a lot of experience investigating strange or inexplicable situations and putting together plans for dealing with them in a hurry. Also he looks good in a suit.
How competent is your character in his/her skill set?: He's quite good at what he does, but modest about it. He works well under pressure, saves the world from aliens on a weekly basis and prides himself on his coffee. He enjoys working on a team and takes pride in being professional and good at his job no matter how odd it happens to be.
Would you like to play out the interview?: I'd be happy to but if its inconvenient or there are time constraints or whatever not doing it works too!
Would you like to play out the skills test?: What I said up there. ^