As far as how replacing the panel would be accomplished, a very simple explanation would be:
The electrician would identify the circuits and shut off the power to the whole box. The wires would be slipped out of the knockouts they enter - usually it's NM wire (Romex) with clamps that slip into the holes and are held in place with a locknut. The whole box would be removed. A new box would be mounted to or in the wall (most likely in) and the appropriate knockouts punched out and the wire clamps and then the wires would be put back into the holes (knockouts). The hot wires would be connected to new, appropriate breakers (a good electrician should attempt to balance the load if it isn't already), the neutrals to the neutral bar, the grounds to the equipment grounding terminal. Then after checking, they should turn all the breakers off, turn the power back on to the whole box, and turn the breakers back on one at a time, checking the circuits as they go.
An electrician may very well pigtail some of the wires to make them reach the new breakers (and this is allowed by the current NEC code book as far as I know).
What I've written is not really instructions, just sort of an example. Basically, a good electrician can replace the breaker box and will know what to do if problems occur (and there can be none or a whole bunch when replacing a whole panel). Always have an electrician do this (don't do it yourself). Also, an electrician should be able to figure out before starting if there are problems like no ground wires, wires too small, etc. (Current code requires ground wires and 12 gauge wiring for "branch circuits").
As far as the dishwasher box, it probably would have been better to put a junction box further away and run either waterproof conduit to the receptacle, or run waterproof cabling to the receptacle. Then you could have just added however much wire you needed from the other side of the junction box to the receptacle. I'm not sure if a junction box in a crawlspace would have been best either if its that hard to get into (there are rules about where you can and can't put a junction box), although you might have been able to fish the wire into a wall further away and put a junction box there (but then it would be in a room - junction boxes are supposed to be "accessible"). There are actually waterproof junction boxes and fittings as well. If a new panel is decided upon, just have the electrician look at the dishwasher stuff too - its a lot cheaper for them to do the work at the same time than to have two separate calls.
By the way, if an inspector is ever called to the property they probably will complain about the FPE panel. I don't know about your area but mine requires inspection before an apartment is allowed to be rented to someone.
Always check with somebody else too, as I'm not an electrician (I'm just interested in electrical).