No, it it's not possible to restore a database from an ldf file. The ldf file would be restored along with the mdf files.
No, it's not redundant as they have two different purposes.
It's important to take full backups, and transaction log backups. Only having a copy of the ldf file doesn't help you restore the database.
As to what a ldf file is for, the ldf is the transaction log. Think of it as a circular buffer that records changes to your database. When you update a row, the change is immediately written to the ldf. At some point in the future (usually less than five minutes), the modified data is written to the mdf file.
If the server crashed or there was a power failure, when SQL starts, it reads the ldf and re-applies (REDO) those changes.
Additionally, if you have a transaction that hasn't been commited and the sever crashes, all changes made by that transaction have to be undone to make the database consistent. The ldf file has that task as well. (UNDO)
I mentioned above that the ldf file is circular. Taking a transaction log backup (.trn) copies out a portion of the ldf file. After a trn file is safely created, sql can reuse that portion of the ldf file. The series of trn backups create a chain that together record every modification made to the database. Of course, if you never took a tranaction log backup, the ldf file would grow and grow and grow.
In a disaster scenario, restoring the full backup gets you a copy of the database as of the time the full backup finished. You can then restore the trn files in order and bring the database current to any point in time including up to the last trn backup.
I'm glossing over some important details, but the gist is the that ldf is a working file that records recent changes to the database. The trn files are copies of parts of the ldf made under the assumption that you will keep then safe so that sql can reuse the space in the ldf and if disaster strikes, you'll have them in an alternate location.
If you did not take a log backup between the full yesterday and your restore today, then no, everything changed since yesterday is gone.
In a best case "accidently deleted data" scenario, the process is to take a log backup to capture the "tail"of the log, restore a full backup of the DB to an alternate location, restore log backups using STOPAT to get to the point right before the accidental delete and copy the data.
If you don't have the room to restore an extra copy, or the accidental deletion is severe enough that you must restore over the existing database, then it is twice as important to backup up the tail of the log.
Best Answer
During a Restore of an existing database, the transaction log will be completely overwritten. If data you needed to retrieve was written after the last known good backup (Full or Transaction) and you've already restored the database then the log is overwritten and the data is lost from an SQL Server perspective.
If a Server or SAN backup is taken, it could be possible to recover a copy of your original .mdf and .ldf file from here although this is generally frowned upon For real recovery you should only use native SQL Server backups but if this is the only option to recover a production database, then you can't be faulted for that but it is not a good solution in its own right. But I would attach as a separate database and run CHECKDBs on it to make sure there is no corruption first.