Why is my wiring a fire hazard when we have circuit breakers?
It's a lot like aviation.  "Little" code violations can stack up and interact to become big ones.   The electrical code is written in blood and ash, which is to say it's less about theory and more about actuarial analysis of accident patterns - pro and con.  For instance blown insulation around knob-and-tube wiring was banned, and is now being un-banned because better data showed it wasn't actually causing fires.  

The dryer circuit sounds like the bog-standard typical 120/240V 30A dedicated circuit: both poles of hot, neutral and (we hope) safety ground.  Safety ground can be retrofitted using any viable path back to a grounded location (not a water pipe).  All 3 (non-ground) conductors appropriate to 30A, that means 10 AWG (rules don't allow you to run the wire at 90C thermal rating, so #12 is right out). 

The washing machine circuit must necessarily be a different circuit from the dryer (since the dryer uses almost 80% of circuit capacity, and also, it's a different voltage). A few washers are designed to plug into partner dryers; these are popular in large complexes because it means one less circuit to the washer/dryer area.  

You're correct that a 15A breaker "should" protect you, because it's small enough to protect wire as small as #14, the smallest permitted in US mains wiring.  However, that assumes everyone behaves properly, and that frequent trips don't bait someone into doing something stupid.   Sort of like the way you can cause accidents by getting on a section with 20 miles of no-passing zone and driving 5 under the speed limit -- it incites stupid people to do stupid things to get around you.  There's no accounting for stupidity in accident statistics. 

For instance, plain overloads cause breakers to trip in thermal mode.  The breaker heats up in theory about as fast as the wall wiring does.  In practice, they cool off a lot faster than in-wall wires - especially if you hit them with canned air.  So if a person  resets the breaker over and over; the breaker cools off during off times, but the wires just keep getting hotter and hotter.

Both washing machine and kitchen countertop circuits are required to be 20A and dedicated.   You have a 15A circuit serving both loads, so you're bound to have overloads -- most kitchen appliances having anything to do with heat are 1500W (12.5 amps) because UL imposes a 1500W limit on such appliances.  That means so much as making toast with the washing machine running makes a trip likely.  Two kitchen appliances also overloads a 20A circuit, but with only a 25% overload (25A vs 20A) you may have enough time to make toast.