I think you misunderstand what factors cause a person to become ritually impure and the dietary laws known as kashrut. The crow/raven is in a class of birds that are "unclean" meaning that they are not suitable for eating. The Torah's list of clean birds is limited to birds who are not birds of prey and those who are not scavengers, like the crow. These we cannot eat. See Lev. 11:13-19; Deut. 14:11-18. Also forbidden to Jews for eating are animals that do not have cloven hooves and do not chew their cud. A dog or cat clearly do not meet the criteria, but there is no prohibition against a Jew owning either as a pet, touching it, or feeding it.
There are many ways to become ritually impure, or tameh, including touching, or being in the same room with, a dead human body (two different kinds of tumah), contact with dead animals not slaughtered in accordance with kosher rituals, contact with certain bodily fluids such as semen or menstrual blood, or, in the case of women, following the birth of a child. See this short and not quite complete list of things that can make one ritually impure. People become ritually impure on a regular basis, and the Torah knows that. The act of being tameh merely prevents one from participating in rites of the ancient Temple and, for those otherwise eligible to do so, being tameh prevented them from eating of food brought on the altar. Most issues of ritual impurity are resolved through purification rituals within 24 hours to two weeks, depending upon the circumstance. In fact, the Torah created a second observance of the Passover festival for persons who were ritually impure when the regular festival came about. Numbers 9:7.
Touching a living bird, even a non-kosher one called "unclean" by the Torah, does not make one tameh on its own. However, the dead animal or pieces of a dead animal) in its mouth could make one tameh. Also, the food the raven would bring would be most likely not kosher, and therefore forbidden to Elijah except in extenuating circumstances (which arguably he was in).
Numerous rabbinic commentators take up your question. Rabbi Yechiel Altshuller (ca 18th century), in his book Mezudath David, takes an alegorical view to these verses, rather than a literal one. God tells Elijah that he will be fed by ravens, which are a species normally known as cruel; if a raven can show Elijah mercy, God is saying, Elijah should take a lesson and have pity on Israel. Rabbi Levi ben Gerson (1288-1344), aka the Ralbag, assumes that in this instance Elijah is not concerned that the food the ravens will bring is kosher because God is sending it to him, using the birds as His messenger. And, some, including Rabbi David Kimchi (1160-1255) interpret the word "orevim" as meaning "merchants" rather than "ravens." Still others take the path of explaining that the ravens brought food from a kosher kitchen on God's command.
As for your question about whether Elijah would be breaking the law by not appearing in Jerusalem for the three pilgrimage festivals violated a law: The law to make the pilgrimage is a positive commandment (i.e. "Thou shalt") which, if a person has the opportunity and the means to do it, he has fulfilled the commandment. Not everyone could do so, especially those who lived in far off nations, although that would not be so true in Elijah's time. But someone who was ill would certainly have an excuse not to go. The Torah and rabbinic commentaries make it clear that this was so as there was a system for Jews abroad to send money to serve as their Torah required half-shekel offering to the Temple via messengers. See, e.g. Jerusalem Talmud, Tractate Shekalim, chapter 1.