In broadcast AM radio it is standard practice to use quarter-wave towers (because half-wave towers are expensive and more of a maintenance headache). These towers require a ground plane to reflect the signal in such a way that it virtually provides an image of the rest of the half-wave that allows for lowest voltage and highest current at the feedpoint. (That's the how and why)
Radials are set 3 degrees apart, so there are 120 of them. Ideally it will be on nice conductive soil, and they are buried just deep enough for good coupling and also so lawn mowers can pass over without causing damage.
The radials are cut to a quarter wavelength plus 5%. It turns out that having the highest conductivity (the wires) provides the best match when it is at resonance. (Every station has a matching network, also). This is in order to provide the image necessary for lowest voltage and highest current at the feedpoint that you would find in a center-fed half-wave antenna.
As they couple more loosely into the ground the actual effective ground plane is much larger than the wires, which helps for ground-wave propagation. Soils vary in their conductivity, so the predictability of providing the resonant wires is the most important factor in impedance matching.
Ground planes are a maintenance issue, and the way they are soldered really matters. Corrosion gets to them and in order to pass my inspections (I did the VIP - the voluntary inspection program that prevents FCC surprise inspections - my reports went into the files at the stations and not to the FCC, and they liked me because I came with a toolbox to fix any deficiencies right then and there). I always wanted to see good solid ground connections at the antenna base, in particular. I always liked it when stations had a good wide low-resistance copper strap at the tower base, so even if some wires get damaged the rest would carry the freight.
Hustler has some excellent references for what they sggest for their 5BTV antennas, and they are similar except that no amateur is going to put out 120 wires. In fact, you can get quite a good ground plane with many fewer wires than that. My personal recommendation is for about 20-40 wires, and of course there are a lot of obstacles in the way of a typical ham vertical antenna installation.
Installation on a rooftop requires the creation of an artificial ground - a counterpoise. It should be tied to earth ground at the transmitter. I have designs for Part 15 LPAM stations that can reach out over a mile with a decent counterpoise of 20 wires. Those must by physical necessity be less than a quarter wavelength; the vertical element is limited by regulation to 10 feet from the ground. (I recommend the Procaster transmitter for this, which is FCC Type-accepted and includes an internal tuner, a bolted-on 9 foot antenna and a 1-foot grounding cable so it can be mounted above the counterpoise.)
Downsloped grounds for vertical antennas can work at about the same length as flat ones, a quarter wavelength plus 5%. Bandwidth is an issue for verticals that must cover an entire band like from 144-148 MHz, so you want the parts to be cut for the middle of the band range you intend to use it for in order to achieve the flattest SWR over the entire range of frequencies. NOAA weather radio antennas should be cut for 162.5 MHz.
So is bigger always better? It depends on your wavelength. VHF signals have little to no ground-wave propagation mode, and the sloped radials help launch the signals skyward. The vertical component holds the launch angle down, of course.
Performance varies slightly between coaxial dipoles and sloped-radial antennas, but not as much as you might think. Similar at UHF.
Beyond that (or even starting at 70CM) you will want to leave omni antennas behind as much as possible. Just ask any police department what their experience was when they shifted from VHF to UHF and they'll tell you it was better before.
Portable radios work best when they have 5/8 wave antennas so they have no need for a ground plane. Quarter-wave antennas in HTs operate under the assumption that the radio body will capacitatively couple to your body to provide a (lossy, loosely-coupled, irregularly-shaped) ground that can reflect the other quarter-wave that is missing from them (the so-called reflected image).
Impedance is a whole different topic, and involves more than just the feedpoint of the antenna, and it runs the gamut, including matching networks in the transmitter or radio shack, feedlines, etc. Magnetrons in microwave ovens come with an integral radiating element which matches the insides of the tube to the waveguide directly. Only the guys in the lab coats would be able to explain impedance in that situation. I will have to defer to others to give the math for all that.
To summarize the question about when ground plane size starts to matter: It always does. Even in a waveguide at 10.5 GHz. Resonance is always best. Compromises are made in the real world, and they all can be made to work to varying degrees. Bigger is not always better.
Impedance matching and resonance is king.
Matching impedance at the antenna is definitely better than matching it at the transmitter unless you want to cut your feed lines to be an odd multiple of a half wave.