Im EU GDPR certified and was interested in the new regulation from BA point of view. You cannot send emails without an unambiguous consent given by your users. So firstly you must define the purpose for collecting the data and than ask the users to opt in (given by choice) for each segment of the service you wish to offer and opt out which should be mandatory. So the site should clearly state “would you like to receive our newsletter” for ex. You may have users who would be happy to visit the site to comment on it but not to receive newsletter…that’s how it should work. Another ex. the statement of this site “by signing up you agree to the basic rules, Terms of service and privacy policy….is not EU GDPR compliant.

But what, for example, if someone had an email like 029893fjkjfka@blah.com? That, in my view, is not something that could identify anyone as an “identifiable natural person”, or could it?

“I don’t see how this can qualify as data collection in any manner, Kristaps.”
GDPR art. 4 (1) – ‘personal data’ means any information relating to an identified or identifiable natural person (‘data subject’); an identifiable natural person is one who can be identified, directly or indirectly, in particular by reference to an identifier such as a name, an identification number, location data, an online identifier or to one or more factors specific to the physical, physiological, genetic, mental, economic, cultural or social identity of that natural person;
GDPR art. 4 (2) – ‘processing’ means any operation or set of operations which is performed on personal data or on sets of personal data, whether or not by automated means, such as collection, recording, organisation, structuring, storage, adaptation or alteration, retrieval, consultation, use, disclosure by transmission, dissemination or otherwise making available, alignment or combination, restriction, erasure or destruction;

While the use cases you both have described are reasonable, there are still a number of genuine reasons why they are not wholly excluded from the scope of the Regulation. At the end of the day you still perform processing of personal information. Please note that this only applies to individuals, and not legal personas or business representatives in their official capacity.
There are 2 general exceptions that would apply, however, they would not completely exclude you from the applicability of GDPR, but would instead change the scope of its applicability.
Under the “legitimate interest” consent exception you can assume that consent is given and that you can lawfully process the email address in order to return a question from your client. It is possible to put most everyday business interactions under the “legitimate interest” basis, but that does not really help you long term when it comes to other concepts under the GDPR.
In terms of processing email with forms and other methods, it is not exactly true, that there is no personal data stored. Even when you forward the data from the form to an SMTP server, it is usually both the server logs (depending on the application, such as WordPress, possibly also application debug logs) and the email infrastructure itself that preserves the personal information for unusually long period of time.
Most of the hacks related to low traffic sites are aimed at this in particular, since most low budget hosting companies will store these files and backups, sometimes for years, and allow full access to all the emails and related information both from logs, backups and other database sources to anyone looking for them.
In essence, it is sometimes not per se negligence, but just bad practices that have flourished over the years due to nobody really caring that this is now quite difficult to come to grasps with.
In short. Processing emails for legitimate reasons is fine, as long as its done properly and with privacy in mind.
Contact forms and naked emails do not automatically exclude you from the GDPR applicability. You should always investigate the procedure, and take the necessary compliance steps.

“1. What if I got rid of all the contact forms and plugins and simply use an email address that people click on to contact someone? Would that qualify as not collecting data?”
I don’t see how this can qualify as data collection in any manner, Kristaps. If I publish my own email address on my site and someone clicks on it or copy it and, then, they send me an email, I am NOT collecting personal information at all. Excuse me but it does nor make sense at all.”
That was exactly my thoughts!!!!! I mean, they are contacting ME, not me contacting THEM. Otherwise, hey, would the law not then need to also extend to your friends (or anyone for that matter) emailing you and you emailing them, if the above scenario is included in the new law?