First, I will assume that when you said, "Developer Suite" you meant version 10.x since this was the last release to have a Developer Suite.
So, sure. You can run those forms on your machine. Simply install Developer Suite 10.1.2 or Application Server 10.1.2 on your local machine.
Forms applications are like Java applications. You need the runtime environment in order to run them. Since an Oracle Forms runtime environment isn't native to any OS, you will need to install one. iDS or iAS both provide what you need to run the application. Also important to understand is that the "X" files (fmx, mmx, plx) are not portable. Meaning, you should not try to move them from the machine on which they were compiled. It can be done across machines that have exactly the same Operating System and version, but generally this is not recommended nor would it be supported by Oracle.
Finally, you said you were new to Oracle Forms. Welcome to the family! You should be aware that Developer Suite is an older product and no longer considered supported. If you are simply trying to learn Oracle Forms, you are probably fine using this version. However if you are working on a project for professional reasons, I would recommend moving to the latest version which is 126.96.36.199. Seems like it only makes sense to learn what's current in my opinion.
Without understand what your end goal is, it will be difficult to make suggestions.
Keep in mind that many concepts have changed from v10 to 11. In 10, there was (in AS) Oracle HTTP Server who talked to OC4J who talked to the Forms runtime. In the case of 10, OHS and OC4J were required for deployment. A client could not directly access OC4J to run a Forms application. In version 11, the family is called Fusion Middleware. In that family you will find things like WebLogic Server, Forms/Reports, and various others. Forms can talk directly to WLS, although using OHS is recommended.
To answer one of your questions directly, YES, you can run your form which was built in v10 in v11. However, in order to do so, you would need to compile it in v11 using the provided compiler. However, once you do this neither the fmb or fmx can be moved back to v10. In other words, the version 10 Builder will not be able to open the fmb again. So keeping copies would be important. That said, I don't recommend doing things this way. If you want to use v11, then use 11. If you want to use 10, then use 10.
Version 11, and Fusion Middleware in general is somewhat more sophisticated and therefore more complex than its predecessors. So, reviewing documentation and having a plan is a good idea.
As I mentioned (maybe indirectly), there is no Application Server (or Developer Suite) product any longer. In order to run Forms/Reports using v11 you need to install at least two pieces: Weblogic Server followed by Forms/Reports. However, as I mentioned, I do not recommend simply downloading these and trying to install them. If you do not properly configure the environment and ensure that you have met all the prereq's, you will certainly have problems installing.
If you want to use the very latest, here are some docs to consider reviewing:
Product Certification Guide for 188.8.131.52. Ensure that you are about to install on a machine that is certified for this product version:
If you don't find the platform you want in this doc, refer to the previous doc, which is here:
Next, refer to the System Requirements Guide. Review carefully the resources, network, and user requirements.
Once you've made it this far, you are ready for the Installation Guide. That doc is found in the Product Documentation Library, along with many other valuable references:
Just in case you decide to move forward without reading the docs, here's are two tips:
Good luck and have fun