to see if the filesystem you're sat in is mounted read-only:

mount |grep $(df .|grep -o "/dev/[^ ]*")

use find . -type d to see if you can traverse the hierarchy, if not then a directory might not have the execute bit. If this is the case let us know and we'll direct the right followup cmd.

use: fuser -m .
this may show a load of process ids and one might be locking your ability to delete stuff. one will be your shell you can look up with
ps |grep PID