Low impact refactoring and code cleaning of sloppy code while waiting for requirements

Question

I inherited an existing code base for a product that is reprehensibly sloppy. The fundamental design is woefully inadequate which unfortunately I can do little about without a complete refactor (HIGH coupling, LOW cohesion, rampant duplication of code, no technical design documentation, integration tests instead of unit tests). The product has a history, high exposure to critical "cash-cow" clients with minimal tolerance for risk, technical debt that will make the Greeks blush, VERY large codebase and complexity, and a battle-weary defeatist approach to bugs by the team before me.

The old team jumped ship to another division so that they have the opportunity to ruin another project. It is very rare that I experience a Technical Incompetency Project Failure as opposed to a Project Management Failure but this is indeed one of those cases.

For the moment I am by myself but I have a lot of time, freedom of decision and future direction and the ability to build a team from scratch to help me.

My question is to gather opinion on low impact refactoring on a project like this when you have some free time during the functional requirements gathering phase. There are thousands of compiler warnings, almost all of them unused imports, unread local variables, absence of type checking and unsafe casts. Code formatting is so unreadable and sloppy that it looks like the coder suffered from Parkinsons disease and couldn't control the amount of times the space bar was pressed on any given line. Further database and file resources are typically opened and never closed safely. Pointless method arguments, duplicate methods that do the same thing, etc..

While I am waiting for requirements for the next feature I have been cleaning low-impact low risk things as I go and wondered if I am wasting my time or doing the right thing. What if the new feature means ripping out code that I spent time on earlier? I am going to start an Agile approach and I understand this is acceptable and normal to constantly refactor during Agile development.

Can you think of any positive or negative impacts of me doing this that you would like to add?