Are unique constraints on the DB necessary?
For me, categorically yes, see Database as a Fotress by Dan Chak from 97 Thinks Every Software Architect Should Know. He says it much better than I could.
Yes, they are. They enforce data integrity at the lowest level.
You might want to change DB content manually(i.e upgrades to new version)
You might forget to check some constrain in your code.
You can look at this like client/server validation. Your program is client, db is server. Mostly client validation is enough, but you must have server validation just in case something goes wrong.
I think a data person would say both are absolutely necessary. Your question assumes that your middle tier application code will be in front of that database now and forever.
The truth is that middle tier applications come and go, but data lives forever.
There's no getting away from length of columns in schema design. I think you're asking if it's a good practice for the middle tier to enforce them. Maybe not, but they're key for the database.