Can we have electronics with charge carriers OTHER than electrons? Ions perhaps? Or other elementary particles with charge?
 A: Depending on your view, there is electronics with other charge carriers. It is commonplace to have semiconductor devices where the relevant carriers are holes!
Furthermore, batteries and electrolysis relies heavily on ions as charge carriers (but hardly count as electronics). I guess genuine electronics with ions will be difficult as charge carrier mobility will be much lower in solid state systems (and high carrier mobility is highly relevant for effective devices).
Furthermore, we are beginning to build electronics, where not the charge, but the spin carries the information ("spintronics"). Some spintronics devices are already in production, namely read-heads for hard disk drives based on the giant magnetoresistance.
Another topic in research is functional replacement for electronics based on optical signals. Signals in optical fibres are already amplified actively without transferring the signal to electrical current (by pumping a dotant, usually Erbium in telecommunication applications, in the fibre, that then emits photons by induced emission when the signal passes). There even are optical transistors.
A: Can we have electronics with charge carriers OTHER than electrons?
Yes, see what Sebastian said above. And see the physicsworld article Taming light at the nanoscale:
"Look around, and you will probably see numerous electronic and optical gadgets, such as mobile phones, personal digital assistants, laptops, TVs and digital cameras. These may all do different things but they have one thing in common: in the electronic circuits that drive these devices, charged particles flow through components and impart power via what is known as the conduction current. But is the motion of charged particles the only current we have available? Those with a good memory for Maxwell’s equations of electromagnetism will remember that in addition to the familiar electric field, E, there is also a displacement field, D..."
You can contrive "electronics" using displacement current rather than conduction current. And get this: displacement current doesn't necessarily involve charge carriers, because "it is not an electric current of moving charges, but a time-varying electric field". The article is to do with "taming light" because light involves a time-varying electric field, because light is in essence alternating displacement current. That's why we talk of vacuum impedance:
$$Z_0 = \frac{E}{H} = \mu_0 c_0 = \sqrt{\frac{\mu_0}{\varepsilon_0}} = \frac{1}{\varepsilon_0 c_0}$$  
It isn't the same thing as impedance in a conduction-current A/C circuit, but it isn't totally unrelated.  
A: You need something that can be conducted along the wire to power electronics; if you were to get protons, rather than spreading from atom to atom you'd just end up changing the element of the atom or splitting it. The closest thing that you can do other than add electrons is chemically charge it, as in replace the batteries.
