I assume you do not want to require the students to own Filemaker to access the tests.
I assume you do not want to have them download a runtime singleuser database, do the test and upload or email the completed test.
I assume you do not want to pay or learn how to use Custom Web Publishing with Filemaker. http://fmwebschool.com
This leaves Instant Web Publishing.
Filemaker Pro, Filemaker Pro Advanced will allow five concurrent (at the same time) users access to the database through a browser.
Filemaker Server Advanced will allow 100 concurrent web browser users
Sharing Your FileMaker Pro file Using Instant Web Publishing
What options do I have to publish data to the web with FileMaker Pro? Updated: Dec 14, 2011
IWP Questions asked here
Your assumptions are correct, at least for the moment.
I will look into the Instant Web Publishing to provide my students access to the tests.
The other part of my question concerned the processing of students' answers and generating reports. I was wondering if there are any pitfalls to be aware of or if there are methods that are considered more efficient in handling this.
Some templates are well constructed, but difficult to decode the logic and naming conventions for a beginner.
I highly recommend the Campus Productivity listed at the end. Good place to start, easy logic and naming.
The White Paper for FMP Novices is useful
Lynda dot com has Filemaker Videos (portions are free)
Free unlocked templates are useful for examining design
Starting Point -
A list of free templates and such, including most of these
A free calendar is available to be bolted onto your database
You can search specific Filemaker sites on this Custom Google Search
Looking at FMP business database demos is useful - some are fully useable
The Excelisys Business Tracker V3.0
Filemaker released a FMP runtime database in 2007 for students called "Campus Productivity Kit" and is available online, notably at CNet
Google "Campus Productivity Kit"