I need to our users to upload files to their own custom folder at least to rename the uploaded files. Is this possible?

Starting from BreezingForms 1.8.5 build 856 you can use so called File and Folder Masks to create dynamic filenames and folders.  

To do so you can replace/extend the "Upload directory" option in the advanced settings of your upload element.

FOLDER MASKS EXAMPLE:

{ff_uploads}/{username}/{date}|

Possible Result:

/path/to/site/media/breezingforms/uploads/admin/2014-07-01/

Possible placeholders:

{userid}

{username}

{name}

{date}

{datetime}

{FIELDNAME:value} => FIELDNAME is the name (not title) of a field of your form and its value is used to change the folder name

FILE MASKS EXAMPLE:

{ff_uploads}/{filemask:_username}{filemask:_separator}{filemask:_datetime}

Possible Result:

/path/to/site/media/breezingforms/uploads/admin_2014-07-01_35:59:59.png

{filemask:FIELDNAME} => FIELDNAME is the name (not title) of a field of your form and its value is used to change the filename

{filemask:_separator}

{filemask:_username}

{filemask:_userid}

{filemask:_name}

{filemask:_datetime}

{filemask:_timestamp}

{filemask:_random}

COMBINED FOLDER AND MASKS EXAMPLE:

{ff_uploads}/{date}|{filemask:_username}

Possible Result:

/path/to/site/media/breezingforms/uploads/2014-07-01/admin.png