I have no option for emissaries or the like.
You have to appoint a companion as minister for that.
I did manage to officially get a truce with the Brychenoig, but they immediately turned around and raided one of my villages, starting the war again (which is ok, I didn't mean to accept peace, it seems if you select the change your mind option it doesn't work.)
If they are allied with any faction that is still in war with you, they will just declare war again (though it doesn't show up as a message) and proceed to fight you.
Also, two lords have defected to my faction but I can't find them anywhere. This may be because I cannot find my court.
You court is the first city/castle you took and declared yourself a king. Your wife and minister are there, as well as lords that are waiting you to accept them to your faction.
How do you pursue diplomatic relations with a faction? So far the only way I have found is to approach the leader of a faction and offer to make peace with them.
Dispatch an emissary from your minister, the emissary himself should be your companion with highest persuasion skill. First you have to "express your goodwill" until that accept you as a king. After that you will have more options, like offer non-aggression pacts or trade agreements.
The situation is untenable, I spend so much time running around taking back and defending my cities I can't do anything else, I'm at war with half the factions in the game. There is no way to get peace!
Perfectly fine situation. Few major alliances fighting all the rest, with only few factions that are in war with less then other 2. And then there you are, a self-proclaimed king with no traditional relations and real right to be amongst kings.
Beginning may be tough, but later this diplomacy system will work out fine. I already had myself accepted as a rightful King by the major factions, my kingdom is almost all britton lands with 6 lords under my command, and I'm in alliance with Mierce (Which is probably the strongest kingdom of all now), Hwicce (the only britton faction that didn't declare war on me after I took Brychenoig and declared myself a king) and Mummain (which is basically Kingdom of Ireland by now, only Ui Neil is still holding). Now whoever declares war on us gets what they deserve.