Electrical – the maximum current a 13Amp twin wall socket can handle

A 15A outlet is rated for 15A, with a 20A pass through. That is to say that the 20A circuit is in continuance throughout the circuit, however the receptacle itself (the contacts) are rated for 15A continuosly. Any single appliance with a 15A plug will not normally draw more than 80% of 15A, or 12A. The total circuit draw (multiple appliances - same circuit) can be 20A before the breaker trips.

A 20A breaker used with 12awg wire can feed multiple 15A outlets, one example is the kitchen. The reasoning is so that today's demanding appliances, which draw more current, can be used with a 20A breaker without the worry of nuisance tripping. If more than a total of 20A were to be drawn from the circuit, the breaker will trip.

It should be noted that any circuit that is intended to be 20A must use a 20A recepticle.

If the appliance were using 15A, it will be safe with the 20A breaker ( @ 80% = 16A). If it were to short, it will trip the 20A breaker just as it would a 15A breaker. A 15A receptacle can take 20A for a short time with no problem. The receptacle is overrated, otherwise it would blow up upon a short. A short circuit in actuality can be hundreds of amps in a very short duration. The breaker and receptacle are rated as "Time overcurrent" meaning it can take a lot of current for extremely short durations, and will trip on lesser currents that occur for a longer time.

An example one can relate to (refer to chart): Joe plugs in two electric heaters in his family room. Everything works fine until 20-100 seconds later the breaker trips! Joe overloaded his 20A circuit, by drawing 40A! The breaker will allow this overload for a short time. If the overload were bigger, say 60A the breaker would trip faster from 10-35 seconds. If the trip was due to a direct short, the breaker will trip Immediately. Breakers actually have a "Load characteristic curve" that you can tell when it will trip in time vs current.

First off, the first electrician is wrong in saying that the use of a crimp-type terminal in house wiring is categorically unsafe. Crimp-type terminals listed under UL 486A for use on solid wire of the given gauge are considered acceptable for use in building wiring, as per UL 486A section 1.1:

These requirements cover pressure wire connectors and soldering lugs for use with copper conductors according to the National Electrical Code, NFPA 70.

and NEC 110.14(A):

Terminals. Connection of conductors to terminal parts shall ensure a thoroughly good connection without damaging the conductors and shall be made by means of pressure connectors (including set-screw type), solder lugs, or splices to flexible leads. Connection by means of wire-binding screws or studs and nuts that have upturned lugs or the equivalent shall be permitted for 10 AWG or smaller conductors.

Terminals for more than one conductor and terminals used to connect aluminum shall be so identified.

However, I would be quite hesitant to call this specific application fully Code-conformant -- garden-variety listed crimp terminals are listed for use with copper wire only, and it would be unlikely that the previous homeowner had access to terminals identified for use with aluminum wiring.

EDIT based on updated info about the devices:

Since the existing devices are copper only, your options go as follows (in order of preference):

1. Find an electrician in your area who is qualified on the Tyco/AMP COPALUM system for pigtailing copper to aluminum -- these have a long, successful field service history, but require a specially qualified electrician to install, and are rather expensive -- you might be able to get the seller to shell out for this, though...
2. Use the AlumiConn connectors with a torque screwdriver for pigtailing. This is something that most electricians can do, or you can DIY if you know your way around electricity and have a torque screwdriver -- the AlumiConn must be torqued to spec to meet its UL listing.
3. Replace all the devices with modern CO/ALR types. This is a last resort as CO/ALR device performance has been historically variable, unfortunately.