Can I carry large sum of cash (>10k EUR) in my hand luggage?

Since you seem to be travelling entirely within the EU you will need to check with the customs offices of the countries you leave, pass through and enter what the arrangements are, since this hasn't been fully harmonised. As you state in the question, it's likely (but not absolutely guaranteed) that you'll need to fill in a national declaration form, but unlikely for there to be any fee involved.

Unless it involves discrimination law, airlines are usually able to make up their own conditions of carriage, so in theory one could ban the carrying of high value monetary instruments, however I'm not aware of any which go beyond "don't do anything illegal, don't put money in your checked baggage and it's at your own risk as much as we can legally say that".

Can I take on-board large sum of cash (more than 10k euros) in my hand luggage?

This may depend on the specific countries involved, bu in general the answer is yes. It is legal to carry as much cash as you want. That DOES NOT MEAN that you are allowed to bring the cash into the country. See the next point:

I know I need to declare this cash at customs but do I need to pay any fee for that?

Typically there is no fee. HOWEVER, you will get questioned and whether you are allowed to bring the cash in and/or enter the country yourself will depend on the quality & credibility of your story and is up the indivudal customs inspector. You better have a good story and some documentation to back it up.

Can the airline have any regulations that would prohibit flying with large sum of cash?

They can, but the typically don't. If in doubt, check with the specific airline.

The point is that you might be allowed to carry the cash. Or not. The salient word here is "might".

The border police is required to ask you both where the money comes from and what you are planning to use it for. As we all know, using cash is a way to sever the trail of money and is often used as part of money laundering. And this means that the border police will be, rightly, very suspicous.

Especially the last few years, as SEPA payments are by "European law" required to work both fast and at low cost within EU+, what would the reason be to carry cash.

Purely, as an example, assume that you are planning to buy a car for 20.000 Euro and the seller requires cash. Your part of the transaction might be fully legal, say saved money. But what happens at the other end? Will the seller rightfully declare the money and pay the necessary taxes? Did the seller acquire the car legally? Will the seller use the cash for illegal purposes?

So, the border police might give you the choice of either board the plane or stay and explain the money. You will of course get a receipt and will be able to get the cash back. But are you willing to take the chance?

As opposed to checked baggage? On your person is better, certainly never in checked baggage. The airlines prefer this too.

10,000 Euro is not a lot of money, neither is 20,000. Perhaps we are just poorer than we think we are. It can be hard to find an ATM in many countries. Is Paris still using GPRS for taxi payments? How a city with such a low skyline gets such poor coverage I will never understand, maybe a lack of towers... Better to have a bit of excess than be caught short. You need to share the joy.

A million may be a problem. Do your homework, but again, not in the checked baggage, never. Do not split it across your companions either, that does not go down well.