Latin is a heavily inflected language, which means the changed endings of words tell you their use or function in a sentence. Word order is flexible in Latin sentences because the endings of words instead of their relative placement reveal their function and meaning.

In the case of "Christus", the "-us" ending is for the nominative case, usually used when the word is the subject of a sentence.

The "-um" ending in "Christum" means the word is in the accusative case, used when the word is the direct object of a transitive verb or the object of a preposition that describes motion or action towards or to, which the preposition "ad" does.

In the case of the word "peccator", it's the same principle, only the way of declining the word is different because it's a different category of noun. Whereas "Christus" is a second declension noun, "peccator" is a third declension noun so it has a different way of changing the word's endings: "peccator" is nominative case and "peccatorem" is accusative case.

To show possession, genitive case, the corresponding words would be "Christi" and "peccatoris", which mean "of Christ" and "of a/the sinner", respectively.

When addressing someone directly, the vocative case is used. For "Christus" that would be "Christe".

The phrases you mention seem to be from a dialogue like in some parts of The Imitation of Christ. "Christus ad peccatorem" would indicate what Christ is saying to the sinner; "Peccator ad Christum" would indicate what the sinner says to Christ.

Dominus is a 2nd declension Latin noun. It's ending follows the conventions of the 2nd declension, which you can learn more about here:
https://www.nationalarchives.gov.uk/latin/stage-1-latin/lessons/lesson-3-second-declension-nouns-to-be/

No, it would not. the endings of the words indicate subject and predicate, not order. You can even say Ad Christum Peccator or Ad Christus Peccatorem, or even Christum Peccator Ad or Christus Peccoratem Ad. Word order isn't important.

Don,

The answer to your most recent inquiry depends on more context.

A specific preposition can also mandate a particular form of the noun as its object.

Some prepositions inherently take accusative objects, some ablative. For prepositions that can accept either form of the object, the meaning of the preposition changes depending on the form used. Frequently, when motion is intended, the accusative form of the noun is used, and when something is static, the ablative is used. There is a pretty good explanation here, with some examples.

https://www.nationalarchives.gov.uk/latin/stage-1-latin/resources/stage-1-latin-grammar-resource/prepositions/

I’m no Latin scholar, but I can’t think of a time where I’ve seen the object of the preposition come before the preposition. I’m pretty sure the object is supposed to follow the preposition.