Working backwards, on your second point, if the FDIC ran out of money, the taxpayer might eventually have to step in, which might not be politically feasible if taxpayers have bail-out fatigue.
"During the savings & loan crisis, the FDIC did not have enough in deposit insurance receipts to pay for the Resolution Trust Corporation wind-down vehicle. It had to get more funding from Congress." [3]
On your first point, the full paragraph provides better context, where they are saying they want to use insurance funds to recapitalize the bank and prevent a bust from happening in the first place. Of course, the bank might need more money, leaving the insurance fund empty, and deposit holders at risk.
"34 The U.K. has also given consideration to the recapitalization process in a scenario in which a G-SIFI’s liabilities do not include much debt issuance at the holding company or parent bank level but instead comprise insured retail deposits held in the operating subsidiaries. Under such a scenario, deposit guarantee schemes may be required to contribute to the recapitalization of the firm, as they may do under the Banking Act in the use of other resolution tools. The proposed RRD also permits such an approach because it allows deposit guarantee scheme funds to be used to support the use of resolution tools, including bail-in, provided that the amount contributed does not exceed what the deposit guarantee scheme would have as a claimant in liquidation if it had made a payout to the insured depositors."
[1] http://www.bloomberg.com/news/2013-01-23/u-s-deposits-post-b...
[2] http://dealbook.nytimes.com/2011/03/22/4-wall-st-banks-still...
[3] http://www.nakedcapitalism.com/2013/03/when-you-werent-looki...
Saying the deposit guarantee scheme may have to contribute means the bank may not have sufficient assets to cover insured deposits and so deposit insurance funds may have to make up the difference. When your house in Staten Island gets razed by Sandy, the insurance company pays up. Similarly, if an insured bank is unable to come up with enough cash to cover insured deposits, the FDIC has to pony up.
The worst case for FDIC insured deposits is you getting a cheque from the FDIC. If JPMorgan Chase goes under with worthless assets and busts the FDIC, you're still in luck since the federal government backstops your deposit insurance, and unlike Cyprus, the U.S. can print dollars.