A question for the native Japanese speakers: what is your interpretation or feeling of 〜ている verbs?

In English, 〜ている is taught as the continuative form (歩いている = walking) and the form for the result of a state change (知っている = know, as in "I know"),
but from what I know about Japanese grammar, my intuition is that this form isn't any different than the use of 〜て as a connective form (座って食べる) and いる, as in to exist.

What do you think of this intuition?
Do you view utterances such as "食べている" as a single word or two words (or maybe even three)?
If so do you feel いる has any semantic meaning or is it purely an auxiliary verb like ます (食べます) to you?

Somewhat related but can there be ambiguity in meaning between a resultant state and continuative action?
For example, can "パンを食べている" mean both "(I) am eating bread" and "There is bread which has been eaten"?