Was the original NYT article's claim that there was a 31.5% fraud rate concerning medicare in 2006 accurate?

tl;dr: No. The NYT article is about claims for "Durable Medical Equipment" that accounted for only 1.7% of all Medicare expenditures in 2006. (i.e., 31.5% of 1.7% is much smaller than 31.5% of 100.0%).

Or, yes, if "concerning medicare" applies only to "Medicare claims submitted by sellers of durable medical equipment." (see also note on "fraud rate" below)

As a result, Medicare did not detect that more than one-third of spending for wheelchairs, oxygen supplies and other medical equipment in its 2006 fiscal year was improper, according to the report. Based on data in other Medicare reports, that would be about $2.8 billion in improper spending. - NYT

The 2006 Medicare expenditure for "Durable Medical Equipment" (DME) was $6.9 billion (1.7% of the $403.7 billion in total Medicare expenditures for 2006) - National Health Expenditures, Table 4. In other words, if the 31.5% in "improper spending" applied to Medicare as a whole then the "fraud" would be around $124.4 billion more than the $2.8 billion stated.

Brief note on what "improper-payment rate" means

The National Health Care Anti-Fraud Association, an organization of about 100 private insurers and public agencies, estimates that some $60 billion (about 3% of total annual health care spending) is lost to fraud every year, but that figure is considered conservative. In 2008, government-wide "improper payments" cost the U.S. Treasury $72 billion, or about 4% of total outlays for the related programs. Of that amount, 50% took the form of reimbursements to providers, medical suppliers, and other Medicare and Medicaid vendors. Medicaid had an estimated improper-payment rate of 10.5%, or $18.6 billion, for the federal share of Medicaid expenditures — the highest rate of any federal program. - source

Note that 50% of the 10.5% "improper-payment rate" was under-payment.

And is the interpretation that the fraud was intentionally hidden from auditors backed up by any evidence?

tl;dr: Not that I could find.

The NYT article does not support Coulter's claim that "Medicare had aggressively hidden the fraud from outside auditors." Statements attributed to lawmakers or their staff are worthless without corroborating facts. (Protections provided by the Speech or Debate Clause allow lawmakers and "congressional staff" to say pretty much anything - factual or otherwise. Coulter's protections flow from U.S. Const. amend. I).

The company that was doing the auditing at the time, AdvanceMed, is still at it and I could not find any evidence they had been investigated (or even given a stern talking to).

Nor could I find any evidence that the episode resulted in even a "shake up" at the Centers for Medicare and Medicaid Services.

Note on "fraud rate"

As mentioned in the other answers and comments, the terms "improper payment rate" or "error rate" and "fraud" are not interchangeable.

From Articuno's answer,

It is important to note that the improper payment rate is not a "fraud rate," but is a measurement of payments that did not meet Medicare requirements. The CERT program cannot label a claim fraudulent. - source

From Black's Law Dictionary, 9th ed.,