tamei tahor ritual purity - What happened to Tohorot?

As I understand it, all of Tohorot has been reduced, in modern practice, to Niddah and some hand washing. The entire structure of transmittable 'uncleanliness' is inoperative. Why is it inoperative? Because there's no Temple, and so no ashes of the parah, to get anyone into a definitively tahor state?

Answer

It's actually almost entirely still operative. People and utensils transmit and accept impurity just like they always did. What has changed is a lack of impetus to purify one's self in most instances. There are essentially three ways of overcoming a given impurity:

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  Metzora's Korbanot: A Metzora is impure and he can only become pure again by bringing certain Korbanot in the Beit haMikdash. This latter impurity is the one that is mostly inoperative because no one can become a Metzora without being declared such by a Kohen and this doesn't happen currently both because we aren't so sure who is a Kohen and because we aren't so sure what shades of white qualify.

  
  


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  Parah Adummah (Red Heifer): One who became impure from a corpse can only become pure through being sprinkled with the ashes of the Red Heifer. We currently don't have any of those ashes available.

  
  
  


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  Mikvah: This works on just about everything else and we still have it. In most cases though it isn't worth using as one will remain impure from the currently unsovable corpse impurity which we assume we all have (a pretty good assumption). It is still utilized to remove a category of impurities called Tumot HaYotz'ot MiGufan (impurities related to bodily discharges) which can be undesirable for other reasons (prohibition of sexual intercourse with a Niddah or Zavah, prohibition on entering the external parts of the Temple Mount for all (with possible exclusion of Keri), possible rabbinic prohibition on learning Torah/praying for a Baal Keri, rabbinic prohibition on Baal Keri Kohanim eating Challah of the Diaspora). Other than that, no one really minds remaining impure from most things, because going to a Mikvah won't change their status practically. (Technically there are different levels of Mikvahs for different things, but they are all possible nowadays.) Without being able to become fully pure and do things like eat Terumah and Challah, offer most Korbanot and enter the inner sections of the Temple Mount, most people are not careful about contracting most impurities. This is not a new phenomenon, see https://judaism.stackexchange.com/a/20280/759 for more.

  
  

The laws that accompany Taharot have a number of applications to other areas of Halacha. Sechach for instance can not be a vessel which would accept Tumah. Produce in Yerushalayim must come in contact with a liquid before having Maaser Sheni seperated from it. Kosher Mikvaot are needed for vessels bought from non-Jews (this is not an impurity concern). Hand washing before bread as you mentioned has some related concerns, including what amount of food requires washing. Male Kohanim are prohibited from contracting corpse-related impurity (this has lots of applications). There are likely some other small points which I am not thinking of now.

In the meantime, by learning the Halachot of Taharot it should be considered as if we kept them.