The first phrase should be read as part of the sentence in the previous verse, Luke 16:20. This is important because it is a clue to the "crime" for which the rich man is to be punished in the story. The beggar desired to be fed, but he was ritually unclean and technically should have removed himself from among people and not begged from them. However, those who were unclear were left food. Seeing him, the rich man could have done this, but under Jewish law it is not a crime to ignore others. The second part of this verse, starting with the "moreover" is clearly a new sentence.  Dogs are mentioned here because they were considered unclean animals under Jewish law.  This verse has two unique words and several uncommon ones. 

The Greek word translated as "and" is used as the conjunction "and", but it also is used to add emphasis ("also"). 

"Desiring  " is a Greek verb that means "to set one's heart upon", "to desire", "to covet," and "to long for."

"To be fed" is from a Greek verb that  means "feed", "feast", "fatten" and "to eat their fill." It is a term most commonly used for cattle.  It is a passive infinitive, so "to be fattened". 

The word translated as "with" means "from" in both location and when referring to a source.  

"The crumbs which fell" is translated from a Greek word that means "to fall" and "to fall down." It is the root word for dozens of Greek terms involving moving from a higher state to a lower one. The form here is an adjective, "fallen" in the form of a noun, "the fallen" or "those droppings". 

The word translated as "from" means "from" in both location and when referring to a source. This is the same word translated as "with" above. 

"The rich man's" is from an adjective that means "rich," and "opulent." It very much has the sense of ostentatiously rich. Here, it is used with an article, "the rich" but unlike English the sense is not a group of people, which would be plural, not singular, but "one who is wealthy." It is possessive, singular, "of the rich one". 

table: "Table" is an uncommon noun for Jesus to use. It means "table", "dining-table", "eating-table", "money changer's table", and "grinding stone". 

This is where a new sentence starts. 

"Moreover" is from two Greek words that mean "but also".   The Greek  "but" here denotes an exception or simple opposition. It is used to emphasize the contrast between things like we use "rather". It is the Greek word "other" like we use "otherwise" or here, "instead".   The Greek word "and" is used as the conjunction "and", but it also is used to add emphasis ("also" or "even). Here the sense of this phrase seems to be "Inse

The Greek word translated as "the dogs", means a "dog". As in many languages, it was also an insult. Specifically, it implied shamelessness and in women and recklessness in men. Dogs were considered unclean animals under Jewish law. They are used here to emphasize Lazarus's unclean state. This is an uncommon word for Jesus to use. 

The word translated as "came" primarily means "to start out" but Christ usually uses it to mean "come" but not always. It indicates movement, especially its beginning, without indicating a direction toward or away from anything, so it works either as "come" or "go," but it is more like our phrase "being underway." Our English word "show up" captures both the "start" and "come" ideas.  Here, it is in the form of an adjective, "showing up". 

There is no "and" here. It is added because the verb above was translated as an active verb instead of an adjective. 

"Licked" is from a Greek verb that Jesus only uses here that means "lick up".

The word translated as "him" is the Greek word commonly translated as third-person pronouns in English.  

"Sores" is from a Greek noun that means "wound", "festering wound", "sore", and "ulcer".   It is only used by Jesus in this verse.