1) use a portal to a related table for logging each item checked out. Look up "portal" in Filemaker help to learn more.
2 and 4) The related table from 1 will fulfill this requirement and also that of 4. You'd set up three tables with this relationship:
Students::StudentID = Student_Equipment::StudentID
Equipment::EquipmentID = Student_Equipment::EquipmentID
Equipment would have one record for each instrument, lock and any other equipment you might check out to them (Uniforms for example?)
A Portal to Student_Equipment on the Equipment layout would list all students that have check out the equipment both in the past and the present. You can sort the records in this portal so that the top row shows who has currently check out that item (and use a status field in Student_Equipment that shows that an Item has been returned.)
3) Put a status field in your Equipment table that shows whether an Item is Out or In.
Define a calculation field in Equipment as: If ( Status = "In" ; EquipmentID )
Set up your value list to list EquipmentID as field 1 and an Equipment Name or description as field 2.
More sophisticated versions of this can be set up that use a relationship and a category field so that you can specify a type of equipment: (Lock, Uniform, instrument...) and then the drop down automatically updates to list only available items from that category.
To learn more about conditional valuee lists:
Forum Tutorial: Custom Value List?
Knowledgebase article: http://help.filemaker.com/app/answers/detail/a_id/5833/kw/conditional%20value%20list
Hierarchical Conditional Value lists: Conditional Value List Question
Feel free to post follow up questions here if those links aren't enough to get your value lists working correctly.