It is possible to slightly change the User-Agent string
Right now the string is something like: "/Satoshi:0.9.1/" or "/Satoshi:0.8.5/"
List of nodes is here https://getaddr.bitnodes.io/nodes/1407404879/?q=United%20States
Just change your bitcoind and set this string to "/Satoshi:0.9.1-Donate-1MyBitcoinAddressForDonations/"
(this can be even configurable option in .conf file)
Everyone will see that the node is really exists and connected
In fact, I do not think that fullnode owners should ask donations. The rest of network is obliged nothing to them.
Right now the string is something like: "/Satoshi:0.9.1/" or "/Satoshi:0.8.5/"
List of nodes is here https://getaddr.bitnodes.io/nodes/1407404879/?q=United%20States
Just change your bitcoind and set this string to "/Satoshi:0.9.1-Donate-1MyBitcoinAddressForDonations/"
(this can be even configurable option in .conf file)
Everyone will see that the node is really exists and connected
In fact, I do not think that fullnode owners should ask donations. The rest of network is obliged nothing to them.
Nope. Anyone can do that, even someone not running a full node. You can't guarantee that they actually have the whole blockchain.
Unfortunately the "good feeling" do not pay the bills. 
The donation is a great idea, it's just the "proof of storage" part that would get tricky and to get people to appreciate the effort, of other doing it, for them.
But you could get a generated list on say "Blockchain" and people could be encouraged to participate, by having a monthly "Lottery" where a portion of the donations, goes towards a single pot, and you get people winning that "pot" each month.
Just a thought.
The donation is a great idea, it's just the "proof of storage" part that would get tricky and to get people to appreciate the effort, of other doing it, for them.
But you could get a generated list on say "Blockchain" and people could be encouraged to participate, by having a monthly "Lottery" where a portion of the donations, goes towards a single pot, and you get people winning that "pot" each month.
Just a thought.
Still not proof. What the OP seems to want is a way to prove, in a way that doesn't require trusting a third party, that an address is associated with a full node. (Note op never said address explicitly, but I don't know where else you'd even start)
Perhaps nodes could have an optional donation address config option. You could connect to a node and ask for its donation address. People who want to donate to nodes would have a piece of software that monitors a node or number of nodes uptime and if it is satisfactory it will request their donation addresses and send a donation.
Wouldn't be too difficult to implement, though I'm not sure how many people would be willing to donate.
Wouldn't be too difficult to implement, though I'm not sure how many people would be willing to donate.
Anyone could pretend to be a full node, but actually just be relaying messages from an actual full node, man-in-the-middle style.
Actually you can receive an indirect income by providing alternate services to Bitcoin network.
Check https://bitcointalk.org/index.php?topic=726826.msg8214655#msg8214655.
This week will be more information about how the node works and potential additional services.
The basic Coinjoin service doesn't require a large bandwidth connection, but if you connect the node to a good internet connection you can provide the blockchain network service and get additional income with a better ROI than a GPU rig miner. You will be able to pay the basic bills (host, internet access, etc) and get extra bitcoins to invest and consolidate the Bitcoin network service.
Check https://bitcointalk.org/index.php?topic=726826.msg8214655#msg8214655.
This week will be more information about how the node works and potential additional services.
The basic Coinjoin service doesn't require a large bandwidth connection, but if you connect the node to a good internet connection you can provide the blockchain network service and get additional income with a better ROI than a GPU rig miner. You will be able to pay the basic bills (host, internet access, etc) and get extra bitcoins to invest and consolidate the Bitcoin network service.
While I like the idea, this doesn't reward someone just for being a full node. They have to do something additional.