Data Files Outgrowing Hardware - Split the File?
1) Yes

2) Yes,

After creating the new files, you'll need to drop and re-create the clustered indexes  of the target tables, moving them on to the new filegroup to move the data:

CREATE CLUSTERED INDEX CIX_YourTable
ON dbo.YourTable(YourClusteringKeyFields)
WITH DROP_EXISTING
ON [filegroup_name]


3) If you can get the DBA to back you up on this solution, I think you'll be OK. Based on your limitations this seems like a perfectly reasonable move - but remember to backup before trying to move the data!