You could take an Enhanced Trait: Toughness power with the Impervious quality, with a sufficiently high Limited modifier specifically calling out you're only immune to damage from Dodge checks that you have succeeded. Note that per p.135 on Partial Modifiers, you will need at least as many ranks in the power increasing Toughness as you would in the Impervious stat (and thus, defeat the intent at being defensive by dodging).
Alternatively, consider the "Impervious" modifier for an Enhanced Trait: Dodge power.
Impervious is primarily intended for Toughness resistance
checks, to handle characters immune to a certain threshold
of damage, but it can be applied to other defenses with the
GM’s permission, to reflect characters with certain reliable
capabilities in terms of resisting particular effects or hazards.
Having a certain degree of ranks in Impervious defense makes the character immune to the effect. For example, a Grenade is a piece of equipment that counts as a Area Damage 5 power. So having 10 ranks in Impervious Dodge would make your character immune to a grenade.
In the end though, the Impervious modifier is what you're looking for. If you want to be able to buy this cheap, then bundle it with Limited so that it only applies when you have succeeded on the test.
It is worth pointing out that the whole "80 ranks in Immunity" to be Immune to Toughness is there to be equal to having Enhanced Toughness 40 and Impervious 40, thus being able to ignore a rank 20 attack. As long as you don't need your character to successfully evade a nuke, you don't need to start from there.
That's not what Healing does.
How do damage and Toughness work in 3e?
Unlike D&D and other systems that use HP, the system Mutants and Masterminds 3e does not represent damage by loss of hit points. Instead it represents damage as a status effect, and the process of taking damage is represented as accumulating degrees of Damage conditions.
When a character is damaged, they make a damage resistance check (d20 + Toughness versus DC, where DC is 15 + the damage rank). Their condition worsens depending on how badly they fail the check, where every 5 below the DC is an additional degree of failure. At 2 degrees, the character is staggered for 1 turn. At 3 degrees, they are staggered until they recover, and at 4 degrees they are incapacitated, and then possibly dying.
Toughness is not equivalent to HP. If one had to compare it to anything in D&D, the Toughness power effect is similar to a saving throw for reduced damage. With the Resilient extra, Toughness would be comparable to damage reduction as well.
So what does Healing do?
By default, the Healing effect can remove Damage conditions from a character, thereby making them less damaged.
The Healing power is very versatile. It can even be used preemptively, so that it would apply the next time the character gets damaged.
You can attach Extras to your Healing power effect, and remove other status effects in addition to removing Damage conditions. Here are some examples:
(The quoted material is from the Heroes Handbook)