(1) & (2) sound like consumer mumbo-jumbo. (3) is common practise. (4) How would you know this, other than from people sprouting (1) & (2)? ;)
Phones, laptops & many other battery-powered devices have carefully thought out integration of the power source, battery, and what's being powered. It's all pretty much a 'solved problem'.
Most of these types of applications are Lithium-based batteries, which do NOT like to be 'float/trickle' charged (unlike lead-acid chemistries, for example, which can and should be left on float charge). As a result, Li-based charge management chips DO stop charging the battery once it's full, and don't start a new recharge cycle until capacity drops several %. Rinse & repeat.
This all happens whether or not the device is actually being used - if the laptop is in operation, all power for the laptop will come from the power-source, but the battery charge management will ALSO still be doing its thing, topping up the battery as required. Only when you disconnect from the power-source will the battery be powering the device.
To your first q1: This switching of power-input to both battery-charger & operation of the actual device is literally switched (with MOSFETs) by the charge controller system. These differ in complexity, and depend, for example, on whether the power-source has the capacity to power the device AND charge the battery at the maximum rate possible ("dynamic power control", e.g TI.com's bq24610 Li-Ion charge controller, & many others like it).
To your q2: This does happen, routinely. It's not "hard", you just have to "intelligently" shunt power around with MOSFET switches appropriately. But, everything has its price, and you'll sometimes see a gadget that can't operate AND be recharged at the same time. There's no inherent reason why not, other than a design decision to reduce complexity or cost or space requirements.
You can see this effect in some gadgets - you put them on charge, charge completes, you leave them on charge for "some time", then take it off, and either (a) you're quickly down to 99% or less, or (b) other times it stays at 100% for much longer - this is the few% battery capacity variance between the battery charge controller waiting until capacity has dropped a bit before starting a new recharge cycle, or if it just happened to complete that recharge cycle when you take it off charge, respectively.