Do I need a dedicated server, if I want students to acces my project via a browser?

Some hosting providers are providing software for you that gives you the basic site in an easy to use drag and drop format, Weebly and Wix, for example. Other providers are providing hosting where you can install any software you want on it.

In Weebly, if you want to upload files, you upload without the ability to control your file section as far as folders go. Everything just dumps into one place and you can browse for your file. If you publish as html output, this will be problematic because it really needs to house all of its files in exactly the same file format as published in its own folder. You can upload just the swf file into a Weebly and because it can run from just that single file it will work. You will not have the mobile friendliness of the html though.

On a server where you chose the software, you do have the ability to organize your file structure and can keep each Captivate project separate. It can be shared, virtual dedicated, or dedicated and work fine with Captivate. I am a teacher as well. Would you like to see how we are using Captivate both inside and outside of our Moodle installation on our server? I can even set up a sandbox course in our Moodle for you to try with your students to see if using Captivate inside of Moodle (Moodle is an open source, free learner management system) would be a good way to go. We also share our Captivate projects, including Captivate cptx files, with other schools and non-profits for free, so you may find some really useful stuff in our library as well. I am off for spring break next week and the week after, so I will have some time to spare to help you. Feel free to contact me at teacher.tammy.moore@gmail.com.

I put demos, captivate projects on our domain all the time. It's not a dedicated server.