When a creature that can see the medusa's eyes starts its turn within 30 ft. of the medusa, the medusa can force it to make a DC 14 Constitution saving throw if the medusa isn't Incapacitated and can see the creature.
A monster with tremorsense can detect and pinpoint the origin of vibrations within a specific radius, provided that monster and the source of the vibrations are in contact with the same ground or substance.
A creature with blindsight can perceive its surroundings without relying on sight, within a specific radius.
You gain a keen sense of smell and an instinct to detect prey. You can see invisible creatures and objects within 10 feet of you, even if you are blinded.
A creature solely using tremorsense is only able to tell the location of the Medusa. Because of this, such a creature would not be able to see her eyes, and therefore would be unaffected by her visage. Even if she stands still, she still has a beating heart or other functions that would create vibrations.
As for Perceiving Sight, if you look through your own blindfold and see her eyes, you can be petrified. Nowhere in the Petrifying Gaze text does it see she must see your eyes, only yourself, which you have not hidden from her.
Perceive the Unseen is essentially the same. If you can stop yourself seeing her eyes, looking only at her auras, you should be fine.
Blindsight is a funny one, but RAW, you don't "see" with it. It states that you instead perceive. Because of this, and because it does specifically say that one must "see" the medusa's eyes in Petrifying Gaze, blind characters are pretty much safe from that attack.
In the Perfect Senses text, we see that the word "see" is explicitly used. If you had this and looked at the Medusa, you're a gonner.
Spells and Attacks.
Most spells require you to see your target. These spells would not work against the Medusa as RAW while you were using solely tremorsense. However, if your DM is nice, they may allow you to cast the spell with disadvantage on your roll or give the Medusa advantage on her save, or a similar homebrew mechanic, just to see if you can aim it in the right place.
As for weapon attacks using solely tremorsense, you would again be at disadvantage, as the rules say that you must be able to see your target.
When you attack a target that you can't see, you have disadvantage on the attack roll. This is true whether you're guessing the target's location or you're targeting a creature you can hear but not see. If the target isn't in the location you targeted, you automatically miss, but the GM typically just says that the attack missed, not whether you guessed the target's location correctly.
HOWEVER, this tweet From Jeremy Crawford states that creatures with blindsight can effectively see. This means that targetting creatures with spells and attacks that require you to see the individual are not at a disadvantage.