I had a postdoc in Belgium. You will find pay/conditions a bad shock after Switzerland. the system is very much about personal contacts so suggest networking via seminar series etc asap. There is one major funding scheme through which you apply for individual fellowships to be held at a host uni - the FNRS https://www.frs-fnrs.be/fr/ - some info is in English. Most others work a stream of very p/t teaching jobs at different universities even 0.1 FT ones - if you don't speak French/flemish you are limited there but check Vesalius College in Brussels also College of Europe and united Nations university where things are done more in English. It will though be quite hard to break into the Belgian university system.
The good news though is if you are open to non university jobs there are many think tanks and organisations in Brussels to work for.

Don't panic. Usually each examiner has to submit a preliminary report about a week before the viva indicating areas of concern and what's good. They meet beforehand and discuss to agree what areas the viva should cover but they only make a decision after the viva, so your viva performance can take items off the list they had at the start for corrections. Don't get too hung up on minor v major corrections - you don't have to take the full amount of time to do them anyway and often examiners will be cautious and offer longer because they know many people have a slump just afterwards and need more time.

See if your UK university has any agreements with France on joint PhDs or 'co-tutelle' - some do and it could solve the problem.

It's 3.5 years of funding. It is unusual to complete a PhD in exactly 3 years by the way most people are closer to 4. It's not like a taught course where everyone ends on the same date.

Social scientist here - I'd suggest you ask your questions directly to the university as what is and isn't included in the different pathways does vary (assuming the information isn't on the DTP website). The DTP my university is part of does offer a 3.5 track for those with a Masters without sufficient methods training so your situation. You have to complete 60 credits of methods modules in the first year. But I know other DTPs that insist on 1+3 in that scenario.
Unlike it seems, from what Rewt says, in the sciences, to maximise your employability as a social scientist, you are best getting as broad a methods training as you can and then specialising in what you need for the thesis. Even if you're not using certain methods you need to be able to understand research that does use them, and having a decent knowledge of quant methods does open up a lot of job possibilities both within and outside academia.

The solution for those I know in that position has been a professional doctorate programme rather than a PhD. But I'd agree with others that your scenario looks unfeasible.

Check the contract - there will probably be a notice period. But personally in your shoes I would absolutely apply for f/t jobs, take anything offered and quit.

How realistic is it that she will get an academic job do you think? Is she getting interviews? I think your reaction is probably different if on paper she's competitive as a candidate and just missing out, compared to someone chasing an increasingly unlikely dream. If it's the former, do you have a sense of what's going wrong and whether say interview coaching might be a worthwhile investment. If it's the latter, I think you need to start stressing that there are good non-academic alternatives, that it's ok to walk away from academia etc. Maybe you could have an 'in an ideal world' what would we want to have in our lives, where would we want to live etc conversation and gradually bring her round to the idea that there are other futures that could be great but don't involve an academic job. Good luck - the uncertainties of such a long job search must be tough on both of you.

Before you do anything re the job, ask your supervisor if there's any way to suspend studies I..e stop the R&R clock on health grounds. It might be possible.

Safest way to make sure you can't sue, while still providing a factually accurate reference.

Everything you write suggests that you should take the job. Research and academia are not for everyone and there's no disgrace in accepting that and walking away to something you think offers more of what you want out of life. I suspect that once you're out of the environment, your regrets will subside as you'll be busy with something else. At another stage in life you may feel differently and want another go but for now I'd quit if I were you.

It's not illegal to give a bad reference - that's a myth. References have to be factually accurate. I would imagine someone will be willing to confirm dates of study and the award of a PhD but given the complaints, I suspect HR advice would be not to go beyond that.

I can understand why you feel upset, but this might not be intentional deception but rather an unfortunate series of events that has engulfed your department. Licences are unpredictable and sometimes regulations change or you get a picky assessor, and with the best will in the world things can go wrong. Also your supervisor might have known generally that the university / faculty was in financial trouble but not the extent to which your department was going to be cut. University leaders keep such matters very close to their chests.
It sounds like he's had to scale back ambition for the group to adapt to the new reality and that includes moulding your project into something that's doable in the circumstances. You have to decide whether it's worth continuing with or whether you want to cut your losses and apply elsewhere.
On the money, if your supervisor is HoD and has to cut 21 million from the department, those are massive cuts. He's probably looking at major redundancies which will gut research groups. He can't in those circumstances be seen to favour his own group for funding particularly if objectively as he's told you, he thinks your project can be done without the funding, when others perhaps cannot. It's absolutely reasonable of you to decide this isn't what was offered and walk away, but I think you are being a bit unfair in your assessment of someone who's probably living through a nightmare right now.

I think most people have felt like this at this stage of the PhD, and I've no real advice other than one final push and you are done. I think in the long run you would feel gutted to have quit so close to the finish. You're doing the right things in breaking it down and doing short concentrated bursts of work. The only other thing I can suggest is rewarding yourself. Could you incentivise this with something concrete, like booking a holiday for the week after you are due to resubmit? Or a day doing something you love, when you've got to a particular point?

Universities have external targets for PhD students submitting on time (usually 4 years but some subjects may differ) and the penalties for missing it can mean losing research council funding for future PhD students. If your department is near the threshold, and your supervisor has a record of students not completing, she probably is feeling under a lot of pressure herself about you finishing on time (especially if you've been funded). I am telling you this as she may be aware that in those circumstances extensions might be hard to get approved by the faculty so is pushing you hard as a result. You need to talk to her, show her a realistic plan of what still needs doing and how long it will take, and jointly strategise how best to move forward regarding an extension request or any loopholes to temporarily stop the deadline clock for you. She may be assuming you have more written that you do so now is the time to be brutally honest.