When will there be no 'mineable' bitcoins left? At today's rate of progression when will the market cap of 21 million be reached? If bitcoin increases at such a rate. In addition to this, is it true that when the market cap is reached, miners will still get transaction fees? If so, how much will these transaction fees be?
Bitcoin will hit its maximum of 21,000,000 coins in approximately the year 2140.

After that, there will be no "reward" for mining, and the only incentive will be transaction fees.   Transaction fees will continue to be set by the market (as they are now), depending on how much people are willing to pay.  

No one can say with certainty what the fees will be at that time.