This answer is partially derived from other answers and comments for this question:

### Answer for question 1

Despite the fact that DOIs _should_ stay operational/accessible, there exist different reasons why they can become inaccessible:
1. **Some publishers might vanish over time.** In this case a DOI authority can be contacted to report the broken DOI.
2. **A publisher might have been merged with another publisher.** In case such a DOI is broken, the publisher or a DOI authority could be notified. Usually, the new publisher can fix it.
3. **Some papers might be revoked after publication.** In this case a DOI authority should be contacted so that the DOI does not link to a URL which e.g. does not exist anymore.
4. **A domain of a publisher might be down temporarily.** This case can be tested by loading other Web pages of the same publisher. If several pages are down at the same time, it's advisable to wait for some hours and then again check the state of the domain. Just some patience might solve this issue.
5. **A technical problem with a specific Website of the publisher's domain might exist.** E.g. the content could have been moved to a different location/address. This can be tested, by checking other Web pages of the same publisher. If they work properly, it's most likely that a specific Website has some issue which needs to be fixed by the publisher. Thus, the publisher should be informed.
6. **Sometimes, when papers were recently published, the DOI might already exist but the paper is not yet accessible.** In case of very fresh papers, just some patience might solve this issue, until the content is uploaded properly.
7. **Some papers may be fake as well or the publisher might provide low quality papers.** In case a DOI is inaccessible, it can be useful to check the trustworthiness of a publisher:
   - Is it a rather unknown publisher?
   - Does the website look suspicious?
   - Are there many typos in the papers or on the publisher's Web domain?
   - Has the publisher published papers only for e.g. 1 or 2 years?
   - Are there several papers with inaccessible DOIs of this publisher?
   - Can you find other inconsistencies?
   - Optional and not recommended in general: Has this paper never been cited by another author or by very few authors (citation count)? This is not a general recommendation, since every fresh paper has to start with a citation count of zero. Furthermore, it is not always perfectly transparent how the citation counts are created by existing systems. Different systems may report different citation counts. Furthermore, the citation count will hardly be completely accurate. If considered, this should only be seen as some small indicator in combination with other criteria.

    The more questions can be answered with yes, the more suspicious it is that the publisher might be fraudulent. It may also happen that this publisher is a "low quality" publisher, which provides low quality papers, without an extended peer review process. In any mentioned case, it is recommended to consider finding alternative papers probably also from other publishers, if possible.
8. **The DOI points to the wrong paper.** If a DOI points to the wrong paper or content, then a DOI authority should be informed.
9. **Some typo might exist in the DOI.** Finally, it is also possible, that there is just some typo in the given DOI.

In case of doubt, one can also simply perform a web search (e.g. Google or Google Scholar, etc.) with a search string containing the title and the authors of a paper to look it up and to check the corresponding DOI. Sometimes there may also exist a different (and operational) DOI for the paper.

### Answer for question 2

https://doi.org and https://crossref.org are two excellent domains for DOI lookup.