right now exist a fee market, the transactions well be done with a fee that depends of supply and demand.

I'm not sure it works exactly like that. I think miners have two incentives from supporting the network right now. One is to discover new blocks and get the reward from it, and the other is collecting the fees from the transactions. I think transactions fees will still be collected by miners even if all bitcoins are mined. So the question actually is, will the fees alone be enough for the miners to keep providing their service.

To be honest I think there will be no problem at all. First that will only happen in more than 100 years from now, so two things probably happened by then. One, is that mining hardware have evolved enough, and it's not so costly tu support the network, second, bitcoin price is so high that the fees are more than enough to keep them active.

The miners will get money from transaction fees you say. But to confirm a transaction they have to put it into a new block. But when all Bitcoins are mined they wont find new blocks. So where do they put the transaction in?

Mining will need to continue in order to confirm transactions. Miners will still need to be paid so their payment will need to come solely from fees. You can imagine that fees will need to be higher than they are now but it will be a gradual process as mining gets harder. It will probably be the end of the century before all Bitcoins are mined so we won’t be around to see that happen anyway.

This has been asked for a million times and already answered many time, please next time use search engines to find your answer. The simple answer here is:
if 21 millions of bitcoin already mined miner will get bitcoin via transaction fee on processing transactions. I do not think that 21 million will be easily done, there is a chart of how much bitcoin per year will be generated (i think it was just an explanation ).

if bitcoin runs out in the mine I think it's going to run as it is today and with a high bitcoin price if the cost depends on the bitcoin price later and we do not need to worry too much now, I think we're in heaven if bitcoin runs out.

Nothing will happen if 21 million Bitcoins are already mined but coins will continue to circulate around the world. Though coins cannot be used anymore if the owner ever lost his wallet and that's why Bitcoin value will continue to in demand because owners and investors are keeping their coins and today only less than half of coins that's been circulated are locked down in the wallets and the time they're gonna spend it is unknown.

After the complete mining of 21 million bitcoins, the bitcoins will start developing more since their number will bow be fixed now the bitcoins will increase in price tremendously, since now supply willbe exhausted people will have to pay more bucks to own it...
They may reach millions easily at that time its not impossible at all, with high price you will have to pay high transaction fees then again forks will come more frequently then because now they will aim at making better right now the focus is divided between making it better and between producing them whatever happens it will be very beneficial for people who are holding bitcoins.

Bitcoin already lost some hundreds thousands in this time if we are calkculating all of the bitcoin that lost due to the various reasons. The miner still run the nodes but they will not receive the more bitcoin reward from the block again. They will depend on the transaction fees to keep their nodes still run to secure the network of the bitcoin itself.
No reward in every block, that means the bitcoin arleady reached the total supply.

Quoting myself:

https://bt.irlbtc.com/view/2164199.msg23368481#msg23368481