why do dropdown menus not save after restarting your computer?

No. It's not the menu selections that are the problem.

It's TestComplete actually finding the objects (in your case dropdown menus) in order to use them.

In order to use an object, it needs to identify it. In order to identify it, it uses properties of the object. Can be a single property, can be multiple. Can force reliance on child objects. Can apply a setting that says "find this object, using these properties, anywhere in the application object tree" which gives you flexibility, but at the cost of performance. (Last one is not a setting I'm a fan of personally) Identification properties will be captures automatically as you record or map objects, but this in no way guarantees the properties used will hold up in the long term. I've always found that manually configuring object maps takes more time initially, but they stay a lot more reliable in the long term.

But, this is all automated testing 101 here. You say you're new to this? You mean TestComplete, or automation in general?

It sounds like you may want to start reading here: https://support.smartbear.com/viewarticle/74488/

And see how you get on.