Is it correct that the filter should be changed at about 5k miles when using synthetic oil, regardless of whether the oil is replaced at that time? If so, is there a best practice to change only the filter (without draining the oil?)
Correct. Dealers and other ASE techs recommend an oil/filter change and tire rotation every 5000 miles. Change every 12 months -- even if you don't have the miles on it. Filters are paper and/or poly blends that get saturated over time. You can go up to 7500 if you drive lightly... your maintenance minder will also tell you about the oil. I change mine every 50% on the minder -- which is around 5000 miles. Everything else is in your owner's manual. Read the fine prints on the minder sections -- they will tell you about when to change other fluids and AT fluid in certain driving conditions (AT and rear diff fluids should be changed at the same time).
Honda's recommended maintenance schedule actually calls for a new oil filter with every other oil change--it even comes up on the maintenance minder that way. I still change it with each oil change, though. Cheap insurance. And a long-established habit. 🙂 (Although I might make an exception if I am traveling--I now have a Fumoto valve on my oil pan. I could empty the old oil straight into an old oil jug and refill with new oil, without having to worry about a mess. I would only do this if I left on a trip within a few thousand of my next oil change.)