Learn Spanish (Latin America): Conque este sinvergüenza tiene mucho descaro.

As a single word, conque is a deductive conjunction that means "so" or "so then." You use it to express a consequence or a deduction based on something you just observed. If it were written as two words (con que), it would mean "with which," and if it had an accent (con qué), it would mean "with what?".

Why do we use the indicative verb tiene after conque?

The conjunction conque is followed by the indicative mood when you are stating a deduction or drawing a logical conclusion, as the speaker is doing here (deducing that this guy has a lot of nerve). It can also be followed by the imperative if you are giving a command resulting from that deduction (like conque vete meaning "so leave"), but it does not trigger the subjunctive.

The word sinvergüenza ends in "-a". Why does the sentence use the masculine demonstrative este instead of esta?

Sinvergüenza is an invariable noun, meaning the word itself doesn't change its ending to match gender. You use el or este for a male scoundrel (este sinvergüenza), and la or esta for a female scoundrel (esta sinvergüenza). In this sentence, the speaker is referring to a man.

What is the relationship between sinvergüenza and in this sentence?