How can I load my slider inside other Joomla extensions?

If you have an extension, which has items, events or products, so anything with a description, you could try out the loadposition or loadmodule Joomla method. The loadposition works in a way, that you can create a module position for example inside an article:

and if you put a module into that position, and choose the Menu Assignment to load on that page, it will load there:

If it doesn't work, make sure, that in your Extensions -> Plugins, the Content - Load Modules is enabled. But it also could be, that it just doesn't work with your extension.

An alternative to "{loadposition xx}" is the "{loadmodule yyy}" variation which is handled by the same plugin.

In this case the plugin looks for the first module who's type matches the string 'yyy'. So, you could load a "mod_login" module by placing {loadmodule login} in your text. If you wish to load a specific instance of a module, because you have more than one login module e.g. titled as login1, login2, etc. you have to use {loadmodule login,login2} for the module titled as login2.