Percentage of mined Dogecoins On http://dogepay.com/ I read that 19% of Dogecoins have been already mined.

Is this measure accurate? (So much mined, wow.)

How long it takes to mine them all?

When all dogecoins have been mined, how can transactions happen?
How long it takes to mine them all?


DogeCoin has a block time target of one block per minute. According to its bitcointalk thread it follows a block reward schedule where the first 100,000 blocks get a random reward between 0 and 1,000,000 dogeCoins and every subsequent 100,000 blocks halves the of the previous reward's maximum.

Beginning with the 600,001st block, the reward will be a constant 10,000 dogeCoins per Block until the maximum of 100 billion dogeCoins is found. 

Since there is a random process underlying, it is not possible to accurately predict which block will provide the last scheduled block reward until the first 600,000 blocks have been mined, however, the first 600,000 blocks should produce an estimated ~98.44 billion dogecoins, so there would be about 156k blocks with a 10,000 doge reward.

At 1 minute per block, 756k blocks would take 525 days, or just shy of 1.5 years.


  Is this measure accurate?


Yes. According to dogechain.info also, 21.45 billion dogecoins (21.45%) have been mined. 


  When all dogecoins have been mined, how can transactions happen?


In short: Mining rewards are composed of both the block reward and transaction fees. The transaction fees will remain, and should suffice to reimburse the miners to continue just as before.
Please refer to Will the proof of work system end when all bitcoins have been mined? for a more comprehensive answer.

Update: Flat 10k reward forever

On February 1st, 2014 the Dogecoin developers announced that Dogecoin would not be capped at 100 billion dogecoins after all, but would continue to have a flat 10k dogecoin reward for each block after Block 600,000.