For example, my developer org was created five years ago. If I install a new app, may it then have some features which will not work properly?
May any old app that I install in my new org and that used to work, not work properly?
All Salesforce Orgs do get automatically updated to the newest release following the schedule, my first developer org which I created in 2012 is on Winter 17 so the current release. To track seasonal releases the Salesforce release blog is useful.
There is a lot of backwards compatibility with the API calls supporting back to version 10 so it is unlikely that any App will no longer work correctly, however it is not completely impossible. For example any old versions of apps would not be lightning compatible but probably would run properly in classic.
Any new app which was installed would work correctly.
Salesforce works really hard to make sure that old apps continue to work release after release. How they do this is by way of The Hammer. Basically, they run all unit tests in all managed packages using the old version, and again using the new version. If the unit tests have different passes/failures in each version, then they investigate the problems and try to make sure things remain compatible. In other words, every app publicly listed on the AppExchange should work when the new release comes out, even if the app in question hasn't been updated in a few years.
As other answers have said, you get new salesforce release automatically. That covers the vast majority a DE org would ever use, so you're generally in pretty good shape there.
What you don't get in old dev orgs is changes that are manually made to the template by salesforce, for example provisioning new (normally) paid features to the DE org template. Knowledge is an example of something that may be missing from a DE org if it was created long enough ago.
There may also be some rare cases where new features that are controlled by a permission that may not be automatically enabled. This is rare, and I don't know of an example off the top of my head, but it can conceivably happen.
If you can't leave your old DE org behind and need new features that are provisioned you can generally get them by logging a case in the partner community, but this doesn't scale beyond a handful of orgs.