Can my players use both 5e-2014 and 5e-2024 characters in the same adventure?
Yes
You can use any rules you like, regardless of what WOTC recommends. However, the 2024 PHB gives you guidance on how to use 2014 sub classes with 2024 classes. The 2024 Players Handbook explains how this works.
Is it possible to run an adventure with characters from both 5e-2014 and 5e-2024 PHB?
Again, you can do anything you like in your game, but yes, this option is supported by the 2024 rules. If I understand correctly, the guidance is to use 2024 classes with 2014 (only) subclasses, meaning the subclass progression may be different from 2014. If a subclass was replaced, then the guidance is to use 2024 version (both class and subclass).
Where might this get weird?
If you have a player that only has the 2014 rules, and the DM is using 2024 rules, the better part of the rules are identical, a few rules have minor changes and a few rules have more significant changes that may cause issues for 2014 (only) players. For example, if your player has a 2014 assassin, and you are using the 2024 rules, how hide and surprise work are completely different, so you may need to follow the guidance and update to the 2024 rules for this class and subclass.
Tools
You didn't ask, but I assume many folks are concerned about the tools, particularly the dndbeyond character builder. According to the Beyond Ruleset Changelog [UPDATE],
Players who only have access to the 2014 Player’s Handbook will maintain their character options, spells, and magical items in their character sheets. Players with access to the 2024 and 2014 digital Player’s Handbooks can select from both sources when creating new characters. Players will not need to rely on Homebrew to use their 2014 player options, including spells and magic items, as recommended in previous changelogs.
Ostensibly this means the character builder will allow you to do any of the three (use 2024 rules, use 2014 sub classes with 2024 classes, or use 2014 rules). This is all just announcements, however, so how the tools will actually work remains to be seen.
They do, however, mention the following.
We are not changing players’ current character sheets, except for relabeling and renaming. Examples include Races to Species, Inspiration to Heroic Inspiration, and Cast Spell to Magic.