Q:

Are URLs in emails indexed by search engines so they become publicly searchable? 

I have read a few questions on here about e-mail clients prefetching URLs in e-mails.  An answer to this seems to be to add a new confirmation page, where the user has to click a button to confirm the desired action. 
But, this answer states the following:

As of Feb 2017 Outlook (https://outlook.live.com/) scans emails
  arriving in your inbox and it sends all found URLs to Bing, to be
  indexed by Bing crawler. 
This effectively makes all one-time use links like
  login/pass-reset/etc useless. 
(Users of my service were complaining that one-time login links don't
  work for some of them and it appeared that BingPreview/1.0b is hitting
  the URL before the user even opens the inbox)
Drupal seems to be experiencing the same problem:
  https://www.drupal.org/node/2828034

My major concern is with this statement:

As of Feb 2017 Outlook (https://outlook.live.com/) scans emails
  arriving in your inbox and it sends all found URLs to Bing, to be
  indexed by Bing crawler. 

If this is the case, any URL in an e-mail meant to confirm an action, e.g. confirming a login,  subscription, or unsubscription, can end up searchable in a search engine, if that's whats meant by indexed in the quote above.  In this case, it's Bing.  Not even a dedicated confirmation page where the user confirms the desired action truly mitigates this. 
Scenario #1
If I email the user a login link with a one-time token in the URL, that URL will end up in Bing.  This token will have a short lifetime, lets say 5 minutes, so I doubt anyone will manage to search on Bing and find the URL before the user clicks it or it expires. 
Scenario #2
The user gets an e-mail with a link to confirm a subscription.  This link is perhaps valid for 24 hours.  This might(?)  be long enough for someone else to stumble over the link on a search engine and accidentally (or on purpose) confirm the subscription on behalf of the user. 
Scenario #2 is not uncommon, it's even best practice to use double opt-in as far as I am aware. 
Scenario #3
Unsubscribe URLs in the bottom of newsletters.  Maybe valid for forever?  You don't want this publicly searchable in an search engine. 
Assume all the one-time confirmation links land on a confirmation page where the user confirms the desired action. 
Is it truly the issue that URLs in e-mails are indexed by search engines, at least Bing?  And will they actually end up publicly searchable?  If not, what is meant by indexed in the quote above? 
I'll add for the sake of completion that I don't think I've had much of a problem with this in my own use of the web, so my gut feeling is that this is unlikely the case. 

A:

Is it truly the issue that URLs in e-mails are indexed by search engines, at least Bing?  

I can't definitely say if they are being indexed or not, only Bing could answer this question, but they are surely being visited, at least with a simple GET request.  I just tested this sending myself a link to a page on my website that logs the requests that are made against it, and indeed I'm seeing a GET coming from 207.46.13.181 (reverse DNS says msnbot-207-46-13-181.search.msn.com), which suggests that an automated program from search.msn.com is crawling the link.  This leads me to believe that yes, they are trying to index the link's content somehow, but it's only my opinion really. 

And will they actually end up publicly searchable?  If not, what is meant by "indexed" in the quote above? 

Well, again, impossible to say unless you work for Bing.  In any case, "indexing" means exactly what you think it does: parsing the content of a page to potentially include it in search results. 

The real question here is: does this somehow represent a security problem or will it compromise my website's functionality? 
It surely has the potential to: if your confirmation/reset/subscription/whatever process only relies on a single GET request with the appropriate GET parameter, then you should definitely revisit the strategy, as it obviously allows anyone to perform the action (even maliciously for example enumerating possible IDs for your GET parameters). 
If the link you are trying to send contains sensible information or can be used to alter important data for an user of your website, then you should at least put it behind a login page only giving access to the interested user.  This way, anyone who wants to access it (including search engines) will be redirected to a login page if not already logged in. 
If the link you are trying to send is just some kind of harmless confirmation link (e.g. subscribe/unsubscribe from a newsletter), then at least use a form inside the web page to do the actual confirmation through a POST request (possibly also using a CSRF token), otherwise you will unequivocally end up with false positives.