Get a dump of the data stored in the Evernote account

It seams Evernote at the time of writing has no support for OATH like many others do. :(

The only way at the moment to give a 3rd party access to your notes is to give it your password. If you give your password to a 3rd party they will have the same access rights to your Evernote as you have. This is true for any other service.

UPDATE: I was wrong! I could link my Evernote account to Greplin without giving them my password. In Evernote it appeared under Settings -> Authorized services. So Evernote does support a form of OATH or can give 3rd parties access to their API without revealing your password to them.

The limits are considerably generous and practically unreachable. I do not think that any one human being in a single lifetime will be actually able to exhaust these limits - not unless you are deliberately taking it as a sport

To answer your question, when the limit is crossed, you will not be able to create any more notes or notebooks or tags. Whether they ask you to delete a few things or create a new account is only a matter of their user interface and are only suggestions ad do not reflect on anything of technical nature. This is the same as reaching the total space limit in email providers, online storage and stuff like that.