Should Data and Indexes be split in separate file groups and files
Improvement (or not) will depend on your storage system. Multiple files can help performance if it increases IOPS and bandwidth with additional HBAs and drives. Multiple filegroups allows more granular placement of database objects. That will generally help in specialized use cases, such as separating sequential and random IO patterns on spinning media.

That said, I don't expect storage tuning to address the symptoms of "massive deadlocks and poor performance". Query and index tuning is the first thing I suggest you look at.