Take a walk through with the home inspector and see what he says. If you are still interested in the house, find a good general contractor (that you trust) and take him on a walk through. The contractor should be able to give you a good estimate on what it will cost to fix any issues, he may also give you some price breaks if you are willing to let him handle all the work.
Just make sure you are comfortable with the contractor before you ask them to go on the walk through, they might be upset if they take time to walk through with you and then you find a new contractor to do the work. Keep in mind, however, that if the contractor starts work and does a sub-par job, you can always fire him and find a new contractor.
Unless you have a really clear idea what to do with one, leave the fire axes to the folks who do know what they are doing with one.
When in doubt, get out. Stuff can be replaced. Dead people can't. Killing yourself by trying to fight a fire is bad enough - if you manage to kill firefighters who go in looking for you, it's worse. If it's small and you can apply an extinguisher without undue risk to yourself, only then try putting it out. Remember that most people are not "burned to death" in a fire - most are asphyxiated.
I shade a bit larger (5 lb., or 2A, 10BC) on what I think is an OK minimum than @mjohns615, but otherwise agree, pretty much. ABC dry chemical is safe to use on any home fire, so you don't have to figure out which extinguisher to grab when you are probably a bit panicky and time is of the essence. I'd rather have more than I need than less, so long as it's not to the point of absurdity. I think the one in the garage/shop is a 10 lb.
The US A/B/C ratings are for a "size of fire that can be put out" for each of the "types" (A being paper/wood, B being oils/fats/grease, C being electrical) and the B/C rating is nearly always 5x higher than the A rating for dry chemical, since dry combustibles have embers which the other two types of fire don't, so they are prone to relighting themselves.
While terribly old fashioned, if you can make space in your life for it, the bucket of sand can be quite effective, and the fire blanket can also. In the kitchen, a box of baking soda and/or salt can also be useful (as is the lid of the pan for a grease fire, if it's handy and you act quickly.) There are also "class K" extinguishers specifically for kitchen grease fires (what we'd ordinarily consider class B) but that's more of a commercial kitchen application than most home kitchens.
Once you have extinguishers, you need to keep an eye on them - most have a gauge to indicate if there's a leak, and if there is a leak, they need to be serviced (mostly for the initially-more-expensive extinguishers) or replaced (most cheap extinguishers.) Likewise, if the extinguisher is used at all, it needs service or replacement (the valves don't seal once powder has gone through them - also the contents are not all there.) You can do the inpection yourself (most homeowners) or you can hire a company (most businesses) to do annual inspections and servicing as needed. If you have a serviceable extinguisher, have it serviced (when needed) by a specialist company.
Best Answer
Swap the lower-left office and the center "bedroom"
The easiest fix for this will likely be to swap the office space in the lower-left of your picture with the "bedroom" in the middle, as the former is large enough to be used as a bedroom and conforms otherwise as well.
As to other approaches...
The problem with the alternative you've floated of using sprinklers to make up for the deficiency you have noted is that the governing Codes that are in use in various jurisdictions disagree about their validity. In your case, your jurisdiction uses a code (an amended version of the California Residential Code) that is based off the International Residential Code (IRC), which in R310, requires that basements, habitable attics, and sleeping rooms have an operable "emergency escape and rescue opening", normally implemented as an operable window of a specified minimum size, and only exempts sprinklered dwellings from that requirement if you're talking about a basement bedroom in a basement that either overall has such an opening in addition to a standard egress means, or two full means of egress (the latter would come into play in a walkout basement, for instance).
However, not all jurisdictions adopt the IRC or a code based on it. Instead, they adopt the NFPA's Life Safety Code alongside a local or national building code; it is even possible for a jurisdiction to adopt both the IRC and the LSC, although such a jurisdiction will likely use local amendments to cross-correlate the two. Under the LSC (NFPA 101) regime, what you describe would be permitted, as while the LSC requires a secondary "means of escape" (again, an operable window or door of a certain minimum size) from all living and sleeping spaces in 24.2.2.1.1 (which'd include your office), it exempts sprinklered dwellings from that provision altogether in 24.2.2.1.2, provided the sprinklers conform with NFPA standards.
Nonetheless, I have no qualms about recommending a fire sprinkler retrofit anyway; California has required fire sprinklers in new residential construction for years now, so you don't have to worry about dealing with daft water utilities or contractors who won't give you the time of day. Your primary challenge will be the probable need to upsize your water service line from the street to provide 15-30GPM at sufficient residual pressure for the system to operate properly, as many residential water services are only 5/8" or 3/4" in diameter and thus lose too much pressure if you try to make them flow that much water; a more practical minimum is 1", with 1.25" preferred, although this does not mean your meter has to be that large, as fire sprinkler water is not metered in most jurisdictions.