I think you've really got all the boxes ticked already on the teaching front, so I think you should probably think in terms of two things.
1) Which job realistically is going to allow you to publish most easily? It's your publications that get you onto long and short-lists for academic posts. If you are over-conscientious on the teaching side, it could be the ft job.
2) Will the admin job add / develop useful skills / experiences for the non-academic job market? However good you are, the odds against getting an academic job are high, so if the f/t job makes you more competitive for plan B, then it needs serious consideration.
For what it's worth, I don't think either job would be held for or against you if the rest of your cv stacks up. If you take the p/t teaching though, think about giving yourself a limit for how long you'll do it for. It's horribly easy to get trapped in that sort of job and not move on.

In most universities you have to give 3 months notice, but the reality is that most people would work out the semester to avoid unnecessary disruption, so as not to burn bridges with the colleagues you're leaving.Traditionally there was a peak period of adverts for lectureships around February / March for a September start date but it seems to have spread out and got later in the last few years - probably as student numbers have been quite volatile for many universities, and people are waiting to get a sense of that before advertising. There are far fewer adverts in the Autumn for lectureships, but research jobs seem to crop up fairly randomly.

I was a civil servant for 5 years between MA and PhD. I do include that - not least as I thought evidence of administrative competence might be viewed as useful and I'd always heard that you shouldn't leave long unexplained gaps on a cv. I certainly don't think it would be held against you, and may be a minor plus point, but I wouldn't let it detract from the main story you are presenting on research and teaching. I wouldn't and don't include the temping and bar work I did for a few months between finishing the PhD and starting a postdoc.

First - everything I say applies to the UK and to research-intensive universities.
@ literarytheorist - sorry you're unemployed and not having much luck with your applications. It's horrible and if it's any consolation many people who have got academic jobs in the last decade have been there. I certainly have. The academic job market has been dire for a long time now.
@ Dunham internships and marks don't count - they don't say anything about someone's ability to make a good academic. So long as you have some teaching experience, it's really publications and research income for those a little further out from the PhD that get you longlisted, What I mean about very similar is that people are all trying to publish in similar places and chasing the same grants, and so you argue about whether a Marie Curie or a Leverhulme postdoc fellowship is more impressive, or whether a minor difference in impact factor makes one person's article stronger than another's. This is why the extra information in the cover letter / statement is useful.
@ both of you, you both agree that the hiring department should inconvenience itself so you don't need to write a cover letter. OK from my perspective there are some issues with this. 1) I get you're not interested in the fairness element, but legally that is really important. Universities don't want to get sued. 2) You don't seem to understand that here in the UK we operate on a tight schedule. Permission to advertise is maybe Feb, we need to interview by May at the latest, as most people we interview will need to give 3 months notice and we need them in September. We also need a senior faculty member on the panel - that really limits possible dates. Adding another stage makes it unviable timewise. 3) You need a cover letter to apply for a basic uni admin job (in my region more competitive than a lectureship) - what makes ECRs so special that it's a terrible burden for them?

Conferences v journals - this is discipline specific. Some subjects e.g. computer science seem to put weight on conference proceedings. Others e.g. social sciences, would say conference presentations are nice to have, but do not count compared with journal articles.
How many? Again it's subject specific - lab sciences where multi-authored articles are the norm, are always going to expect more than subjects where single-authored articles are more the norm. It's also going to vary depending on whether you are applying to teaching or research-intensive institutions. A good rule of thumb is to look at the most recently hired academics cvs and see what they would have had at that stage.

I am a member of staff and recently served on a selection committee and I can assure you that if you are a competitive candidate then that letter will be read. An average UK lectureship in my social science discipline will get c.120 applications. Maybe 50% will be uncompetitive as they lack a completed PhD and/or REFable publications (i.e. a university press book or articles in top quartile journals - my employer's expectations) and so are easily knocked out. The other 50% do have those things but largely have very similar cvs, so that's when we read the cover letters / statements carefully to look for the things I mentioned in my last post. We cannot waste time interviewing someone who can't articulate where their research is going next for example. It would lengthen the timeline of a job application process considerably to only demand a cover letter after an initial sift, which given we are normally working to a tight timeline is tricky, and would open up all sorts of cans of worms about fairness, I suspect. What do you do if a really good candidate is on holiday and so misses the cover letter deadline for example? It's easier if there's one deadline for everything.

If you want to see a real waste of time, look at the North American system and at what they demand applicants for academic jobs send in. I applied for a job in the US that wanted a cv, cover letter, three writing samples, a teaching statement, a research statement, three reference letters (sent separately by each referee in a particular format), transcripts from all degrees, certified copies of my degree certificates and two equal opps forms, all sent hard copy.

Just wanted to say, don't worry about other people's reactions - I think you'd be surprised how many PhD students suspend studies for a bit. Illnesses both physical and mental, personal crises, financial problems etc all can cause people to suspend studies. One thing is for sure though if you are working 8am to 2am, it's no wonder your work is suffering regardless of your illness- no-one can be productive like that. Ann Jolie is absolutely right, and if your supervisors are aware, it's no wonder they are keen for you to take a break. How about proposing a compromise i.e. that you will take a few weeks off, that you'll commit to working healthier hours etc.

OK
1) there are HR rules to ensure compliance with equal opportunities legislation - that means some of the stuff is needed to ensure equality of treatment. Those annoying online forms for example - that's HR.
2) Most of the candidates applying have very similar cvs. A covering letter allows you space to make a case for why your publications have merit etc, and also most importantly gives a sense for the committee of whether you have a research strategy moving forward, and whether you have a clue about the expectations of the role.

According to the 2014 REF, Newcastle's research in English is stronger than Oxford's at the moment.

http://www.postgraduateforum.com/thread-34211 thought this thread might be interesting reading for you. In addition to what I wrote then, I can only add that the education department where I am has really suffered from losing a lot of PGCE places, so one thing to consider is how viable the department offering the EdD is, given it takes a good few years part-time. I get the impression some have suffered more than others.

What has probably happened is that one or more of their successful candidates has turned down the funding. Because ESRC funding is so competitive, many people apply for a few DTCs and the best may well get several offers. Thus rather than lose the funding, they can go back to their other candidates. What your chances are now depend on how many departments are being offered the chance to do this, so it's a guessing game, but at least you're in with a shot.

Would just doing the dissertation give you the subject knowledge you'd need to actually teach that subject? I wouldn't have thought so as it's so specialised. Maybe it's just this region but when FE jobs are advertised here, it seems like the extras they look for are ability to teach literacy and numeracy - might that be a better training track to investigate?

The more good social science methods training you can get during the masters, the easier the transition to the PhD will be, so I'd see what each programme offers in that area to help decide.
I don't think the two years at Sciences-Po is necessarily a deal-breaker if you definitely want to do a PhD in France. My understanding of the College of Europe programme is that it is viewed as a professional masters (M1 in the French system) rather than the research masters (M2 in France) that grants access to a PhD. So if you want to do a PhD in France, you might find that you'd need to do another year anyway.

Depends on the subject and university - in my university staff are not allowed to accept students without either a 2:1 or a merit at Masters as experience has showed that students who have struggled with earlier degrees are unlikely to complete a PhD. The posts on here also suggest that lab sciences are more open to poorer marks than other subjects.