Tá áthas orm a fheicim thú imeacht a dhéanamh.
This is an area of learning Irish/Gaeilge that is very difficult for native English speakers, because it requires us to change somewhat our way of thinking about what is (or is not) a verb.
You need to start thinking of prepositional pronouns, not so much as verbs per se, but as being verbal (that is, being verb-like). Seano explained this to me once, saying, in so many words, that Irish is based much more on prepositions than on verbs, whereas English is based rather strictly on verbs. So it is not so much that an implied verb exists in these usages as that the verbal sense of the prepositional pronoun needs to be understood. In connection with that, we also have to accecpt a certain amount of "fluidity" for the definition (English) that we would apply to any given preposition because their uses do not always follow the strict ways they would be used in English.
In your first sentence, I don't know if the grammar "casann tú" would be objectionable, but I suspect it would be unusual for Irish. Even in English, we might be inclined to say, "I would like to meet with you (leat)" rather than "...meet you." Saying "meet you" in this case implies not having previously met before, so assuming "with you" seems more correct to me anyway. Second, I think that the "you" in your adjusted version would become a direct object, and thus should mutate to "thú". But using "leat" solves both problems.
For #2, "glaoch ort" would literally be "call on you" (rather than "call you", and the idiom implies a personal visit as opposed to a telephone call. I am not sure what context is implied within the sentence as given, but even if a telephone call is meant, I think the use of "ort" will be preferred. Again, thinking of "ort" as having a verb-like quality, means that the call is being transferred onto you as the object of the action.
#3: "Slán" does mean "health", but it has come to be used as the common form of "good-bye". Even the English "good-bye" implies "travel well" or "stay well" at its essence. The "libh" is the plural of "leat" so it means you are saying it to a group who is leaving you, and they are replying to you "slán agat" as the second person singular. Were you with your family, the reply would be "slán agaibh". This is the proper form of saying good-bye when people are departing from one another. The person who is leaving says to the one who is staying, "Slán agat," health remain at you, and the one staying says to the one leaving, "Slán leat" health go with you. Again, plurals may be applied as appropriate. It is also appropriate to simply say, "Slán" with no other embelishment.
4. The use of "cuirfidh" seems to me to be a grammatical way of "shunning the passive voice". The verb bí is almost always considered passive in nature, because it merely denotes existence. "Cuir" has a meaning of "causing something to happen" so a more literal interpretation might be "I will with a purpose call on you tonight." Or, "I will cause myself to call on you tonight." Unusual constructs in English, I grant you, but it puts the impact of action on the will to do something in the future.
I have tried to explain the principles surrounding the use of prepositional pronouns in a verb-like usage, but the specific uses of which prepositional pronouns for which idioms is something you simply learn over time. There is a very good lesson on many of these in the Is Feidir Liom lessons online, which are free, and very good to go through. They can be found at www.isfeidirliom.ie and I highly recommend them. Lesson 6 deals with idiomatic uses of do, ar, ag and le, and gives a very good start to this topic, although there are many other ways that they are used. The title of the program, Is Feidir Liom, means, of course, I Can, and shows one of the many ways that prepositional pronouns are used with the copula, and that opens up a whole 'nother branch of discussion on this topic.
So, I hope these thoughts are helpful to you. Keep striving to understand it, and eventually all these things will fall into place......