No, SQL 2014 is not supported on Windows 11. SQL 2014 is in extended support, which means that Microsoft provides security fixes, but not feature enhancements like support for new operating systems.
It still perfectly possible to install SQL 2014 on Windows 11 - I have instances for all versions from SQL 2008 and up running on my Windows 11 machine.
Beware, though, that there is a known issue with Windows reporting disk sector sizes for some disks that SQL Server cannot cope with. Workarounds exist for SQL 2019, but not for SQL 2014. But if you don't have a disk of the problematic type, this is not an issue for you.
As for SQLLocalDB, it is possible to run multiple versions of SQLLocalDB side-by-side. When you create an instance, you can say:
sqllocaldb create FourTeen 12.0
That is, you can specify a version number to create an older instance.
Any help in tricking or enabling 2014 to work with 2019 without requiring source code updates for old apps would be appreciated if supported on Win11
Not sure what you mean here. Generally, there should be no need to adapt application code for a new version of SQL Server, least of all if you keep the compatibility level at 120. But even if you go for 150, the chances that you will be subject to a breaking change is fairly small. (There are a few subtle ones.)
What is more probable is that you will see changes in performance, to the better or to the worse. (The former is more likely, but you are more likely to note the latter). Again, this is if you go to compat level 150. With compat level 120, you should not see any changes in performance.