Permitting issues and building code vary by local jurisdiction somewhat, but by and large in Washington most places substantively adopt the IRC (and related codes) with some minor changes in their local building code. For the state it is the 2012 IRC (http://app.leg.wa.gov/wac/default.aspx?cite=51).
See also state amendments: https://fortress.wa.gov/ga/apps/SBCC/
Reading the IRC can be daunting. It is enormous and abstruse and hard to search and very poorly written. When you just need simple stuff and not in depth technical requirements there is a great little book called Code Check. My version is based on older version of IRC, but probably still the same for vast majority of stuff. Anyways when I turn to reroofing section of Code Check it tells me that up to 2 layers is acceptable, anymore will require a tearoff (with some limited exceptions). That has been code for years and years, for your contractor to not know that would be a significant red flag. And of course as Jon mentioned also subject to roof decking, structure, ventilation, etc...being adequate.
For further clarification then you can then check Burien Building Code for any amendments they may have adopted to the IRC. Or just call the planning department.
http://www.burienwa.gov/index.aspx?nid=667
Reroof permit in Burien is $106. Ask you contractor for copies of this permit, and all permits he pulled for the project. And while you're at it go on L&I and verify his license and workers comp status. When you can, be there for permit inspections - you'll learn a lot.
For the size of that house a tearoff and reroof should be cheap. Less than 5k. Another option would be to get a roof cert. Though I think they are kind of a scam.
K&T wiring is allowed. I'm not sure I know where to point you that says "K&T is allowed!", I can't point you to anywhere it says it is not allowed either. Main thing is get an electrician to check it out an give you the all clear. An easy check for you regarding your contractor - check the receptacles. Ungrounded receptacles need to be 2 prong or if GFCI receptacle need to have a little sticker on them saying "ungrounded".
Electrical rewire cost is hard to say without seeing the property. Sounds like from your post you already have an updated panel which is good. Rewiring after completing the rehab is generally going to be a bit of a nightmare just because usually it also entails drywall repair, paint, etc.. so I would avoid it if at all possible. Next buyer might not be so sensitive. Also arcfaults, if they aren't already required in Burien will be very soon.
@Nghi Le