There is a previous thread here, but you are right, some mission statement pinned at the top of the forum would be useful. If someone with the right to pin posts comes along, please pin this post.
To quickly go through your list:
1. What’s the CoE? Purpose, mission.
It is somewhat organic, and might have evolved over time. See these two posts to get a good idea:
post 1: [Request for Comment] Council of Elders Governance
post 2: Questions about the Council of Elders
2. What’s the current list of elders? (updateable)
Currently Arbitrage, JRB, Themicon, Uuazed, Surajp, bor and Aventurine.
3. What are the incentives and responsibilities of being an elder?
Personal incentives (not monetary - see [Community Input] CoE Salary).
Recent new development that is an incentive: councilmembers get travel and expenses (food, hotel) paid when attending the “Around the World” meetups. [Proposal] Around the World with Numeratis
Main responsibility is to spend the NMR we got handed by Numer.ai in responsible ways that benefit the numer.ai community. What/how is up to the council, as to avoid making everything a all-community discussion. As a precaution, that spending is done by a multisig wallet, so that at least 4 councilmembers sign off on any decisions involving spending.
4. What is the procedure and schedule to elect and de-elect elders?
Election happens by public vote - see these posts:
First round nominations: Elder Nominations and Nomination Accepting
voting round 1: Elder Voting (Election #1)
one-candidate-only no voting round 2: Nominations for open Council of Elders seat (Dec 9 through Dec 16, 2021)
voting round 3: [Vote] Elder Voting (Election #3)
The only procedure for de-election right now is by a vote of 4/7 council members. There have been votes for term limits, but to be honest, nobody in the CoE or in the community seems to be keeping track of that right now.
post 1: [Community Input] CoE Term Limit
post 2: [Community Input] CoE Term Length
5. What’s the funding? (ideally, include a list of funding events and funding use reports)
this is the CoE ethereum address: https://etherscan.io/address/0xf58b7c28daf13926329ef0c74fa3f7258f5a9131#internaltx
There has been 1 year-in-review post after the first year. Having these more often is a good idea.
post 1: Council of Elders-The First Year in Review
On our previous chat platform (rocketchat) we had an impromptu list of things done that you could consider post 2: 
10:39 AM
The report is planned, but you are right that the council is quiet. But in short: We are multi-signing off on transactions (like numerbay, and a few others), there was a monthly newsletter (that is looking for a new author), we keep an eye on the liquidity pool on uniswap (and switched to one that is cheaper for users), and not the least - there have been a few physical meetups organized with help of the CoE. London and New York last year, for example
What hasn’t happened is the CoE judging and funding new ideas, in part because few ideas have been put forward (nor by CoE members nor by the community). Things have been quiet on that front
ia_ai_Joe
12:48 PM
One BIG reason is the continuous support for community meetups. I will share more info about the Tokyo meetup in the coming days
ResTrading
1:01 PM
Another MEDIUM-SIZED reason is the continuous support for NumerBay, which I think really benefitted the fund especially in recent weeks
Aventurine
8:01 PM
Gm, As we have had no takers in continuing the newsletter in my stead I am willing to take it back and start writing again. I am thinking about changing over to mirror as well for a more decentralized platform for writing. I will look into the mechanics of the change over. As for the weekly CoE twitter spaces our viewership dove off a cliff week by week as many members moved back into full time working status away from home. I would like to propose we change this to a monthly space and we can promote it in the weeks leading up and also invite Numerai team and members of the community to participate. I am very excited to work with ia_ai_Joe on the Tokyo meet up in early apr which I think can reignite more participation across the board. Numerbay has consistently been a great utility for participants and having liquidity pools on uniswap has brought in some nice fee collection to fund initiatives. I think a fireside would be great to do as well in the coming weeks
Bonus question (doesn’t need to be included in the summary): Is CoE self-sufficient now?
Numerbay is not making the CoE money. We pay Restrading for the development.
The CoE does earn some money by providing liquidity on uniswap. The council’s pool is at the low fee of 0.30%, from which the council made a decent chunk of NMR and ETH:
CoE liquidity pool: revert