The basic answer is: you can use an On Scene Load action in the Scene Inspector set to Pause Timeline with the Main Timeline.
To be a bit pedantic on terminology, scenes contain timelines, and those are which play. Every scene has a main timeline which will automatically play when the scene loads. Scene can also contain other timelines you create, which do not autoplay and need to be started by an action.
Therefore in your case the simplest solution may be to not use the Main Timeline at all, since your animations seem fairly independent and would be on separate timelines.
It isn’t clear from your description if you are using Symbols or not.
If you are not using symbols at all, you would create specific animations on alternate timelines, and then create actions to call Start Timeline for those.
If you are using Symbols, then yes, you can create named Custom Behaviors that are set to start the specific timelines in those symbols. You can then use the Trigger Custom Behavior action and enter in the name of that behavior to call it.
One note about Symbols is that they can contain multiple timelines just like Scenes, but also have a Main Timeline. The symbol’s main timeline tends to be used a lot since the point of a symbol itself is to isolate animations, but often they don’t inside the symbol you don’t have the need to run things concurrently. Unlike a scene, a symbol’s main timeline doesn’t inherently autoplay, but Hype will set it up so that it has this behavior. Therefore, if you want to disable autoplaying a Symbol’s main timeline, instead of adding a pause timeline action on symbol load, you can delete the Symbol Action to Start Timeline that is in the Properties area of the symbol element. (If it is a persistent symbol then you would remove the symbol’s On Symbol Load action that is set to start the main timeline).