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We just completed a similar process. Many thanks to @pyridine for posting all the details of your experience - you saved us a lot hassle! (more on that later!)
Background:  I am the child of a US expat living in Sweden. In April, I applied for a job in the US (a very good job with a "famous" US employer).

Our timeline:
 
May 1: receive employment letter, with start date July 1. May 2: email US embassy asking for instructions how to apply for exceptional circumstance DCF at Stockholm embassy. May 2: (half an hour later) embassy replies, telling us to mail in I-130, a copy of the bio page of petitioner's passport, completed G-325As with passport-style photos for both petitioner and beneficiary, as well as a letter explaining our claim to exceptional circumstance (short relocation for a job) and evidence for this. The embassy also informed us that all this will be forwarded to USCIS London. May 3:  Mail in the required documents. May 14: Email embassy asking if they'd gotten our docs. May 15: Embassy replies, confirming that they'd received our mail, and that they'd just sent a follow-up enquiry to USCIS London to find out what was happening. May 16: Embassy tells us that USCIS has given permission to do DCF, and that we need to send in a money order of $535 for the I-130 filing fee and a copy of the bio page of the beneficiary's passport. May 18: Embassy receives money order and copy of passport. May 21: Embassy informs my wife that they've approved the I-130, and sends her the checklist for the interview. May 24: We inform embassy that we've collected the required documents and that the medical has been scheduled for June 7. May 25: Embassy says that June 12 is available for an interview, if it is suitable for us. June 12: Interview at embassy in Stockholm. Everything goes well. Wife gets 221(g) pending medical results. In the embassy checklist, under the instructions for the I-864, the embassy writes "If the U.S. Petitioner has no current U.S. income, he/she will also need to either need to provide proof of assets or find a financial joint sponsor who completes an additional Affidavit of Support with supporting documents."
Thanks to @pyridine, we knew about this at the beginning of May, which allowed us to make preparations! Since we are quite young (mid-to-late twenties), we didn't have the assets directly available. However, my future employer gave me a large advance, which, combined with our savings was more than enough assets.
We made up 2 AoS (the first counting this advance as income-which was more than enough, the second counting it as an asset). The embassy didn't want the income one.
The Stockholm embassy has been really helpful throughout the whole process! All our emails have been quickly answered.
 

Providing they approve the exceptional circumstances, the processing of the I-130 should go quickly (since there isn't a pile of them that the embassy has to go through). It took us about 1-2 weeks to get exceptional circumstances + I-130 approved.
 
You then have to get the medical + schedule the interview for the beneficiary at the embassy - this can take some time, depending on how many free times the embassy has for interview, etc.
 
Overall, from the thread about exceptional circumstance filings, I think 1-2 months seems standard.
 
 

We went through this process a little more than a year ago. Answering the OP's questions:
 
1) No travel to London is needed. The embassy in Stockholm just has to get permission from London USCIS to let them process the I-130.
 
2) Yes, that is what we sent.
 
3) No idea how they make this call. My letter outlining the exceptional circumstances read
 
4) Yes, it is listed as such in the memo for exceptional circumstances DCF.
 
5) I guess the only disadvantage is that if you get married and they don't let you do DCF, then you have to send the I-130 to Chicago and get a CR-1 visa. If I'm not mistaken, this takes longer than the K-1 visa which you could apply for now.
 
Two things to keep in mind:
1) If you get married, you will need the personbevis from skatteverket as the marriage certificate. It can take a few weeks for your marriage to get entered into the system.
 
2) They don't let you use your future income when meeting the affidavit of support requirements.
 
A job offer is literally listed as one of the exceptional circumstances on the I-130 page:
 

As you probably have seen, on the site for exceptional DCF they give a pretty good explanation of the process (they include the email addresses at the bottom of the page, so I guess that is what you do first). I sent a short email to the embassy explaining that I would like to do exceptional circumstances DCF due to short time for job relocation. They then asked me to post them the listed items to the embassy.
 
They list 
as one of the things to include when applying for DCF, so it would be good to get some sort of letter from the employer with the details of the position. 
 
It's hard to know whether they will approve or not. There are probably a few different factors involved, including what type of job it is. They also want a completed copy of the I-130 to look at before they decide whether to actually accept and process it, so I suspect that the ease of approving the I-130 also plays a role.
 
It's also good to start thinking about the I-864 (Affidavit of support). There's a high probability that you won't be able to use your current or future income for it, so either having assets or joint sponsors ready is probably a good idea!
 
I found this post/thread very useful 
 
https://www.visajourney.com/forums/topic/662470-visa-interview-update-dcf-with-exceptional-circumstances-stockholm/?tab=comments#comment-9065201
 
Good luck!

You're fine. The beneficiary's birth certificate isn't needed for the I-130 petition*. It will be needed later (for the visa).
 
*provided that the beneficiary isn't in the USA already.

Your household size is 2. You should leave part 5 no. 3 blank (or zero). This is for people not sponsored in the affidavit (and you spouse IS sponsored). 

Well, you could *try* an exceptional circumstance DCF using the "Short notice of position relocation" example - see the filing instructions for the I-130: https://www.uscis.gov/i-130. If you get permission to do this, the process is a lot quicker-your wife could take the job, and you'd be able to join her in a month or two. 
 
Apparently it is quite difficult to get permission to do DCF from Canada (so far using google, I found one case of this working from Canada on VJ - back in 2014). See below for a thread discussing this
 
On the other hand, since it only takes a few´days for them to decide whether to let you do this or not, there is no harm in trying (you would potentially only lose a few days time if they say no, and you have to send the I-130 to the Chicago lockbox).
 
It would also be another datapoint for VJers interested in this issue.

It's true that there don't seem to be many posts about it, but that could be because not many people try it. However, it worked for us (will post full timeline shortly) and for pretty much everyone else who have tried (that I could find in the posts about it). The OP was asking for the fastest way to get a visa, and DCF with exceptional circumstances appears to be the fastest way (1-2 months).  Note that it is not an expedite, and USCIS clearly state on the page with filing instructions for the I-130 that short-term relocation for job reasons can be a reason to permit DCF at a non-DCF embassy.
 
People who ask for information should get correct info, without (too much) guesswork about what will work or not.