Stack and heap – dynamic allocation question

The basic difference between stack and heap is the life cycle of the values.

Stack values only exist within the scope of the function they are created in. Once it returns, they are discarded.
Heap values however exist on the heap. They are created at some point in time, and destructed at another (either by GC or manually, depending on the language/runtime).

Now Java only stores primitives on the stack. This keeps the stack small and helps keeping individual stack frames small, thus allowing more nested calls.
Objects are created on the heap, and only references (which in turn are primitives) are passed around on the stack.

So if you create an object, it is put on the heap, with all the variables that belong to it, so that it can persist after the function call returns.

While sometimes I think it would be nice to be able to create variables on the fly without generating those pesky "notices", I can often see the advantages of not being able to do so.

I think the biggest benefit to explicitly declaring variables, or checking that variables are set, is code readability. The isset in your example indicates to the person reading the code that $some_variable may not be set. While this doesn't necessarily save much time (as they'll see that it isn't set when they read the notice), it provides the reader with more information, which is always beneficial.