When your feet rub the floor you acquire a small electric charge due to the triboelectric effect. The wiki page explains the mechanism:
"After coming into contact, a chemical bond is formed between some parts of the two surfaces, called adhesion, and charges move from one material to the other to equalize their electrochemical potential. This is what creates the net charge imbalance between the objects. When separated, some of the bonded atoms have a tendency to keep extra electrons, and some a tendency to give them away, though the imbalance will be partially destroyed by tunneling or electrical breakdown (usually corona discharge). In addition, some materials may exchange ions of differing mobility, or exchange charged fragments of larger molecules."
When you bring your finger near a conductor, the electrons on the conductor will relocate due to the electric field from your finger. For example, if you are negatively charged then the electrons on the surface of the conductor will move away from your finger, leading to a local positive charge which attracts the electrons on your finger. If the charge on your finger is large enough, and if your finger is close enough to the conductor, the local electric field can overcome the dielectric strength of the air resulting in the partial ionization of the air. The electrical resistance of the air will fall substantially, allowing current to flow quickly between your finger and the conductor. You feel the shock.
If you bring your finger near an insulator like wood, the electrons on the insulator won't relocate like they did on the conductor. The electric field will be smaller and will probably not cause the air to break down. When you touch the insulator, the charge imbalance will (at least partially) neutralize. This will not happen as quickly as with the conductor though because the resistivity of the insulator will not permit as high a current per unit area, so you won't feel a shock. The insulator will acquire a charge in the area you touched.
Since the insulator does not readily permit the flow of charge, it may not fully neutralize your charge imbalance, so when you go on to touch a conductor you may still feel a shock, albeit weaker because some of your charge has been dissipated.
That should cover 1 and 2. For 3, there is still current flowing (which explains the sparks you see) but since the contact area between your fingers and the key is much larger than the approximate cross sectional area of the spark the current per unit area is much smaller and you don't even feel it flow from your fingers, through the key, into the doorknob.
Number 4: without seeing your travel mug I'd say RedGrittyBrick is probably right when he commented "...your travel-mug is painted as a decorative effect or lacquered to preserve it's shiny finish and prevent corrosion. These layers act as an insulator. You only feel the discharge if it is concentrated in a tiny spot as a spark that leaps across a small air gap from your skin to a conducting surface (e.g. a small scratch in the travel-mug's coating where the underlying metal is exposed, the poured water creates a path to complete the circuit through which the static charge is discharged)."
It is worth noting that pure water is a terrible conductor of electricity. The Z-machine uses 2.3 million liters of deionized water as part of its insulation.
Here is a paper exploring the electrical properties of skin. On pg 848 the authors describe skin's current response to rectangular pulses between $5V$ and $60V$. They observe "in this range of voltages the skin behaves as a nonohmic, nonlinear system, with a conductance that increases with increased voltage. At the beginning of a pulse, the current dropped to a minimal value and then slowly increased."