Making calls to the HIBP email address search API requires a key which is rate limited to one request per 1,500ms. There's a full blog post on why here, this page allows you to either purchase one for a single month, on a recurring subscription charged monthly or manage an existing subscription (i.e. cancel it). There's a US$3.50 per month fee, the reasons for which are explained in the aforementioned blog post. If you've already purchased a key, you'll be able to manage it after verifying you have access to the email address you wish to use (you'll receive a unique link to that address).
At this stage API keys are only available to purchase on a monthly basis. There are cases where annual would make life easier for people (for example, within a corporate environment), please vote for the User Voice idea and you'll be notified if this is implemented in the future.
API keys are rate limited to reduce the opportunity for abuse. At this time, the rate limit is constant for all API users and cannot be increased, although it is possible to purchase multiple API keys under different email addresses. If you'd like to see the option for higher rate limits in the future, vote for this User Voice idea and you'll be notified if it's implemented.
The entire API key ecosystem is tied into Stripe for payment fulfilment. Currently there is no facility to purchase keys by any means other than credit card, for example via PayPal or Bitcoin.
The API key is a $3.50 per month self-service purchase which will automatically send a receipt following payment fulfilment. There are no quotations, no formal documents, no signatures and no negotiations. See the API page for full documentation.
No. If you reach out and ask for a free key for any reason, you won't get a reply.