Here's some good information on "ci" : http://blogs.transparent.com/italian/tricky-little-words-%E2%80%9Cci%E2%80%9D/
This may be wrong but then some of the Italian speakers here will correct me!
Some verbs have a reflexive version. vedere is one of them.
You get the reflexive version by chopping off the last letter of the infinitive and sticking si on the end. So vedersi.
You conjugate the present tense of the reflexive version just like the base version except you stick in the reflexive pronoun. So noi vediamo un cane (we see a dog) but *noi ci vediamo" (we see ourselves - ie each other).
Since you don't have to use the subject pronoun you put ci vediamo. Then fra poco means "in a bit" so ci vediamo fra poco.
Finally, and here I am pretty unsure, instead of putting the reflexive pronoun before the verb you can tack it on the end. Thus vediamoci fra poco.
But do most Italian verbs have a reflexive variant? (Edit A lot more than in English including some where there is no obvious reflexive meaning in English eg ricordarsi remember)
"Vediamoci" is imperative.
"Noi ci vediamo" is normal present tense.
Many verbs can be reflexive, you're right. It's like for you saying myself.
"Io mangio una mela" I eat an apple
"Io mi mangio una mela" I eat an apple myself
I think mainly intransitive verbs cannot have a reflexive variant.
Intransitive verbs are verbs like "dormire" (to sleep), which have no direct object.
(But you can form some strange things similar to reflexive verbs with those as well.)
(Keep on researching, Peter, I will do my best to help. But coming out with a long list of stuff won't help anybody and will take me a lot of time...)
(If you can check online the correlation between reflexive and transitive verbs, I think there should be part of the key to your question.) (When you find something, let me know!)