Most of your questions can be answered "yes", but be careful with Schottky vs. (ultra) fast diodes:
"Can I just buy the higher voltage and current rated Schottky diode and sub them for the lower rated ones?"
Should Work.
"Is the main reason for getting the exact match the cost and ability to fit?"
Some issues like the diode's capacitance have an influence on switching speed and, as a consequence, switching losses, i.e. heat generated in the diode. Try to find at least a similar device. Also, pay attention to peak values for current spikes and the like. Average current is one thing, peak current and power handling capabilities are another.
"Do I only need to match package/voltage/current?"
This is what designers do when looking for second source parts. In most cases, you will be good to go.
"Are Schottky and fast recovery the same?"
Usually, fast recovery is a label for p-intrinsic-n diodes, i.e. Si diodes, that are trimmed towards blocking fast upon being reverse biased. Schottky diodes are, really, always as fast as can be and should not be replaced with slower Si diodes, even when they're labeled "fast". The general rule is: Don't replace Schottky with Si diodes.
"Can I sub an ultra-fast for a fast recovery?"
Using an ultra-fast Si diode in place of a fast Si diode should work as far as switching losses go, but the faster switching action might cause worse electro-magnetic emissions.
"And... at the moment I need a Schottky that is in the TO220 package. Can I use two axial leads and just wire the cathodes together?"
Diodes in three-pin TO220 packages are really just two diodes. They are, however, very similar. When connected in parallel, they will share the load really well. Different packages are also often a hint towards different peak current/power handling capabilities. And, of course, thermal properties will be different. A TO220 has its own little heat sink even when mounted in free air; axial diodes don't have this nice feature.
There are different SPICE parameters..
For example, the NXP model has TT = 3.48E-9
Central Semiconductor has TT = 2.88E-9
Vishay 1N4148W (T154D chip) has TT = 4E-9
None of those looks much like 20ns, but I didn't look at every maker of 1N4148s.
In any case, you may find this Matlab page useful.
Datasheets do not typically provide values for TT and τ. Therefore the Diode block provides an alternative parameterization in terms of Peak reverse current, Irrm and Reverse recovery time, trr. Equivalent values for TT and τ are calculated from these values, plus information on the initial forward current and rate of change of current used in the test circuit when measuring Irrm and trr.
Best Answer
Well, the wording is definitely marketing. But no, they don't mean so fast you won't care, they mean they actually don't have a reverse recovery time. And yes, these diodes are fundamentally different, but they exist and you've probably even used one.
They're called Schottky diodes. Though it's sort of marketing spin to say they have zero recovery time. This implies they are able to do something instantaneously. But it isn't that they have zero recovery time, it's that Schottky diodes don't have reverse recovery charge or recovery time at all. Those terms are not applicable to them and have no meaning in regard to schottky diodes. Schottky junctions do not switch on or off, and something that isn't a switch obviously can't have a switching time. Charge doesn't impact the behavior of the junction, and there is nothing to recover from in the first place, so 'zero recovery diode' is just fancy marketing speak for Schottky diodes.
Schottky diodes are fundamentally different, and are constructed of a metal-semiconductor junction, as opposed to a semiconductor-semiconductor junction like PN junction ('silicon') diodes. PN junctions actually turn on and become conductive in both directions, and take time to turn off, which is, of course, the reverse recovery time.
Metal-semiconductor junctions do not switch on or off, they don't do anything. Their behavior is simply a property of the junction itself. Due to the chemical interface of the metal and semiconductor, the center of the semiconductor's band gap (the gap separating the valance band and conduction band) is 'pinned' to the fermi energy of the metal (since electrons will occupy all possible energy states in a conductor, the surface of this 'electron sea' above which no higher energy states exist to be filled is called the fermi energy).
It's helpful to visualize the metal as a bucket of water and the fermi energy is simply how tall the bucket is. The semiconductor as two pipes, one above the other, with a fixed height in between them. The lower pipe we can ignore, it's the valance pipe and is below the height of the bucket. When we bring the pipes up to the bucket, the center of that gap between the top and bottom pipe is 'pinned' to the height of the bucket. This is called fermi pinning. Why fermi penning occurs is some Seriously Hard Physics that I'll leave up to the reader to discover on their own, if they wish to.
Since the center of the gap is at the same level as the bucket, the top pipe is a bit above the height of the bucket. Now, if you pump water into the top pipe, it can freely pour out of the end into the bucket, like a waterfall. But you'll never get the water in the bucket to go into the pipe above it, because the water will just over flow over the sides of the bucket, it will never gain enough height to reach the pipe.
This is a very gross simplification and takes a lot of artistic license, but that's the gist of a Schottky diode. The energy needed for electrons to pass from metal to semiconductor is not reachable, though a few electrons fly off via thermionic emission (which we see as reverse leakage current. And yes, I mean thermionic emission like in vacuum tubes). The energy needed for electrons to move from semiconductor to metal is easily obtained, and so we see an exponential VI curve. But Schottky diodes actually don't conduct at all in one direction (except by indirect thermionic emission due to vacuum gaps in the mismatched crystal lattices where the metal and semiconductor interface, but this is due to the vacuum, not true ohmic/galvanic conduction), but conduct readily in the opposite direction.
If you can find a meaning for 'recovery' and 'recovery charge' in any of that, let me know. I sure can't. Small signal Schottky's do not have recovery current, nor do they have reverse recovery charge, but many datasheets will incorrectly refer to capacitive charge stored that will discharge back out (like any other charge stored in parasitics) as 'reverse recovery charge'. But this is unavoidable and not caused by the junction itself, but a side effect and fact of life. No current ever conducts across the junction, and the stored charge doesn't need to be removed to prevent reverse currents from flowing. The charge is just something there, but it does not play any role in the function of the diode.
So when a small signal Schottky says it has a 100ps recovery time, that's actually not true. There is some parasitic capacitance that has an RC time constant much shorter than 100ps (or maybe is simply 100ps) but this is a parasitic effect that would exist regardless. It is not caused by the diode itself. In fact, the diode package of small signal diodes is the main contributor to this incorrectly named 'recovery time'.
Now, larger Schottky diodes have more significant capacitance, but again, it is fundamentally different and not recovery. It is an unavoidable parasitic capacitance discharging out, but no reverse current ever actually conducts across the Schottky barrier. Just equal but opposite amounts of charge on either side is leaving both sides at once, but it was already there. Just like any other capacitor.
HOWEVER, the largest Schottky diodes with higher reverse voltages (>50V is the rule of thumb I've heard but I don't really know for sure) require a guard ring to shape the electric field gradient so as to not cause dielectric breakdown in the barrier. This adds significant capacitance, and worse, creates a parasitic PN junction diode that WILL have reverse recovery time and charge. But this is accidental.
Silicon Carbide Schottky diodes have fantastic breakdown properties, and so they are true Schottky junctions and obtain the high voltages you see without the use of any guard ring or other parasitic structures, so yes, they really are 'zero' recovery time diodes. There is no recovery charge, and no recovery time, and no parasitic PN junction diode that can get turned on. But that's just what every small signal Schottky diode has always been. It IS very impressive and an amazing technological development, but from using silicon carbide. SiC FETs and Schottky devices are just bonkers. In performance and price. Hopefully the price part will change though with time, then they'll just be awesome.