Can a clause beginning "I assume" be made into a question by intonation?

Solution 1:

Yes, clauses that start with "I assume" can be turned into questions by inflection.

Usage questions need to be answered by reference to usage, so here are a couple of instances in print:

• I assume you know the story, then? -The Old You by Louise Voss

• I assume you know by now that he died last Thursday? -The Safest Place You Know by Mark Winkler

Each instance is followed by a reply to the question.

The intent of the construct is, as your correspondent noted, to seek confirmation that the stated assumption is correct. There is an expectation of an affirmative answer. A negative answer would typically be surprising to the asker.