Liquid water has a few interesting properties that we need to discuss before answering your questions.
1) It has a high specific heat compared to most things on Earth. Specific heat is a measure of how much energy it takes to heat or cool a fixed amount of the material and has SI units of $\frac{J}{gK}$. Water's specific heat is about $4.2\frac{J}{gK}$ while wine's, at $14\%$ alcohol by volume, is about $4\frac{J}{gK}$.
2) It also has a pretty high thermal conductivity. Thermal conductivity is a measure of how fast energy moves from the warm side to the cold side of an object, and it has units of $\frac{W}{mK}$ (remember, Watts are Joules per second). It turns out this is a super complicated calculation in reality (materials have different conductivities in different directions), but this is a good approximation for this problem.
3) Thermal contact between objects is also quite complicated. Broadly, objects thermally interact by radiating, conducting and convecting (forced and natural). 3a) Radiation happens all the time, and it's how the Sun gets heat to the Earth - all objects emit photons in every direction at a wavelength consistent with the their temperature. Radiation is related to surface area. 3b) Conduction is when two objects are in physical contact and they share energy at their interface. Conduction is also related to the surface area. 3c) Convection is when a fluid moves over the surface of a solid (or when a gas moves over the surface of a liquid). If a solid has a vertical interface with a fluid, the boundary layer of the fluid will move up if the solid is hot and down if it's cold which is called natural convection. Forced convection is when currents in the fluid move over the surface (as with a fan, a breeze or a flow). Convection scales with the surface area of the interface.
All three of these are pretty complicated which is why you've received so many answers that feel like educated guesses. Last, until the whole bottle reaches thermal equilibrium, the freezer will be colder than the glass which is colder than the wine.
A) Wine bottle on rack vs. wine bottle in water: With the tap water at room temperature, the free bottle will be cooler because water has a very high specific heat that insulates the wet bottle. If you got a thermally conductive container with less water, but more of the bottle's surface in contact with the water, you would cool the wet bottle faster, but the geometry of these things is really important, and that is the wrong bowl.
B) Getting water colder than freezing typically changes it to ice. You can lower the freezing point a little by adding alcohol, or a lot by adding table salt. The thermal conductivity of ice is half that of water, so I'll assume that we salted the bowl, and then the wet wine would cool much faster because while the interfaces are the same, the water can accept more heat without changing temperature as much as the air.
C) "Cooling wine is tricky": Not sure what you mean by tricky - you put it in thermal contact with something colder. Glass insulates the wine, but it has a much lower heat capacity and it's a not that thermally conductive. The only way to get the wine cooler than the glass is to take the chilled bottle out of the freezer and leave it in a warmer environment.
D) How freezers work: Freezers pressurize a fluid, expose it to the room temperature and then allow the fluid to de-pressurize it in thermal contact with the inside of the freezer. Feel a bike tire while you let a lot of the air out, and you'll see what I mean. There's typically a fan that pushes air over the pipes/radiator and into the main chamber because it cools the contents more quickly. I leave it as an exercise to decide which of the mechanisms for heat transfer apply here.