Learn English – What does “cut the odds” mean

Henry's answer is exactly right. Here is confirmation:

A clearer, more complete use of "cutting" comes later in the same paragraph.

He wasn't willing to cut a subsidy for heating bills.

This illustrates that "cutting," as used in your example, had an object (parts of the proposed budget), and was not just used as an expression meaning something else.

The sentence in your question might be paraphrased in this way, for the sake of this explanation: "As the government programs fell and the tens of billions in savings piled up during the budget debate in the House last week, freshman Republican Frank Guinta joined fellow congressmen in reducing spending and removing parts of the proposed budget.

crossed signals simply means there was a mistake. As the paragraph says there was a mix-up in communications. As @Fraser adds - "crossed signals" means specifically a mix-up in communications, it is that specific type of mistake that the expression refers too.

"the beer is on us" means "they are paying for the beer". Better rush to Denver on Thursday ;)

If you tell someone "This one's on me" then it means you are paying for that or rather treating someone else to that.