Zacchaeus, -- The name "Zacharrius" is Hebrew, but the Greek form is of a subject, different than the Greek  form of address. Jesus uses this name only here.

make haste, -- (WF) "Make haste" is  a Greek verb that means "set going", "urge on", and "hasten." This is the only time Jesus uses this verb. This is not an active verb but a participle, a verbal adjective, ""moving quickly." 

and -- (IW) These is no "and" here. It is added because the previous participle was  changed from an adjective form to an active verb in translation. 

come down -- The word translated as "come down" means to come or go down, dismount, and related ideas.

for  --The word translated as "for" introduces a reason or explanation so "because" and, in questions, "why." However, since this word always appears in the second position, it is more like an aside remark like, "consequently" or "as a cause". 

to day -- The Greek word translated as "for to day" is an adverb that means "for today" and "on this day."

I -- This is from the first-person, singular form of the verb. It is in the form of an object rather than a subject because it is the subject of an infinitive, which in Greek takes an objective subject. 

must -- (CW)  The Greek verb translated as "must" is a special verb that means "it/he/she needed," and "there was a need." It is not a helping verb like we have in English but an active verb. It is always singular, 3rd person, past tense. Its form is fixed and, when it refers to a verb, that verb's form is an infinitive like we say "he needed to go," but the subject is not separate from the verb.  

abide -- (WF) The word translated as "abide" has more of a sense of to "stay" or "remain," not necessarily dwelling in a place, though the KJV often treats it that way. This is not an active verb but an infinitive.  

at --- The word translated as "at" means "in", "within", "with," or "among."

thy -- The word translated as "thy" is the possessive form of the second person pronoun. It follows "house" so, "of yours".

missing "the/this"  -- (MW) The untranslated word is the Greek definite article," the," which usually precedes a noun or changes the word it precedes (adjective, infinitive, participle, etc.) to act like a noun. The Greek article is much closer to our demonstrative pronouns ("this," "that," "these," and "those"). See this article for more. 

house -- The Greek word translated as "house," is any dwelling place but not exclusively a separate house. It means the household or clan that lives in the building as well.  It has an article so the sense is "the house" or "that house".