The answer broadly is yes, but there are a ton of caveats.
I used to work with applying Nickel-Titanium alloys (usually known as Nitinol) as actuators in the biomedical field. Before going any further I'll relay the actual advice I was given by an applications engineer from a leading supplier of specialist metals: "If you can find any other way to achieve your goal without using Nitinol, you should do it. [...] Often we find that the models are unreliable and we just have to try it and see if it works".
I still harbour a burning hatred for this damnable metal. I'm convinced it hates me. I have no idea what I did to offend it.
Back to your question: Nickel-Titanium (in the right proportions) is a shape-memory-allow (SMA), however in this application the SMA property is not being used directly, rather the Nitinol's superelasticity is what gives the wheels their resiliency to deformation.
Here's a video showing Nitinol's SMA properties:
Wikipedia has a really good definition of Nitinol and it's SMA and superelastic properties:
Shape memory is the ability of nitinol to undergo deformation at one temperature, then recover its original, undeformed shape upon heating above its "transformation temperature". Superelasticity occurs at a narrow temperature range just above its transformation temperature; in this case, no heating is necessary to cause the undeformed shape to recover, and the material exhibits enormous elasticity, some 10-30 times that of ordinary metal.
Nitinol is an intermetallic compound, which gives it a highly regular crystalline structure compared to standard alloys. The crystal structure takes on two separate forms at different temperatures, martinsitic at low temperatures, and austenitic at high temperature. The ability to switch between crystalline states is what gives the alloys it's shape memory properties (I'm oversimplifying, but this is close enough).
The image below shows the Nitinol hysteresis loop, again, Wikipedia has the best explanation:
There are four transition temperatures associated to the austenite-to-martensite and martensite-to-austenite transformations. Starting from full austenite, martensite begins to form as the alloy is cooled to the so-called martensite start temperature, or Ms, and the temperature at which the transformation is complete is called the martensite finish temperature, or Mf. When the alloy is fully martensite and is subjected to heating, austenite starts to form at the austenite start temperature, As, and finishes at the austenite finish temperature, Af.
The key thing is that the temperature at which the shape-memory and superelasticity occur can be controlled by precisely controlling the ratio of Nickel to Titanium in the alloy, and thus shifting the point of martinsitic transformation.
This means that it is possible to create a version of Nitnol that exhibits superelasticity at room temperature, or even at 100 degrees below zero. Thus, no heating is needed to reset the wheels as it could be said that they are in a constant state of "resetting" if the Mf temperature is set correctly.
All that said, the temperature swings on Mars are huge, and Nitinol's superelastic window is only ~40 degrees above the Af temperature. How the designers plan to maintain the superelastic properties over the ~150 degree temperature swings on Mars is unclear. Maybe a thermal management system would be used, or alternatively the rover could be operated only when the temperature would allow optimum conditions for the wheels.
For the other part of your question:
Mars 2020 is intended to be a sister robot to Curiosity with the only main difference being the instruments. This far through the mission campaign it's probably too late to change major components like wheels, especially when the TRL (Technology Readiness Level) for these new wheels is probably only around 5-6 (technology demonstrator) at the moment.