Wow that's terrible!

Does the no-item-in-the-build-queue situation occur if the previous item was just completed by normal production, or even by another chop on the same turn?
No, not really.
If you have a build item in the build queue that can receive Hammers (i.e. not Wealth, not Research, not Commerce) then you should get the Hammers.
When you CAPTURE a City in Vanilla, you are not prompted to select a build item and therefore you need to remember to do it manually. However, when you COMPLETE a build item, you are correctly prompted to select a new build item immediately--so, for completing a build item, you do not have to worry. That said, I have not played with the "minimize pop-ups" game option, which supposedly allows you to delay your build queue selections until the end of the turn, so I have no clue what happens when that game option is in use (i.e. that's a hint of something else that you could test).
Another Chop on the same turn would not matter, either, since you do not actually "complete" a build item until after you end your turn.
There could, be, of course, another related issue to Chopping too many Forests at once--and that is having "excessive" overflow Hammers (more than twice the cost of the build item being obtained in Hammers). Those excess overflow Hammers are discarded. So, yeah, sometimes it helps to stagger your Forest Chopping across different turns, for this other reason of potentially accidentally having "way too many Hammers" and having the game say to you: "look, I'll let some of them overflow from this build item to the next one, but not THAT many--the rest, I will just have to get rid of for you."
I thought that unused hammers were saved up and applied to a later build - is that how it works post-vanilla?
Overflow Hammers--i.e. Hammers above and beyond the cost of a build item--are also stored in Vanilla, too. So, in most "normal cases," things work as you would expect.
If you're going to make a report about this info, then you had best plan to test this stuff out. Therefore, you might as well be prepared to use the World Builder to help speed up the process--for example, adding a stack of Workers to one square lets you instant-Chop a Forest, while adding a stack of Military Units next to the City of an AI that you have already declared war on will let you instant-capture a City. The goal would be to look at each issue on each of Vanilla, Warlords, and BtS and report the subtle differences of each game version.
There are some differences that you could probably ignore, such as which Civics a Leader has across different game versions (sometimes they change), since that info is easy enough to find quickly in each game. However, the more subtle differences, such as changes to the benefits of a Unique Unit across game versions (the Aztec's Jaguar is one such example) are not something that a player would think to look up in the Civilopedia, thus I think that it would be good to capture this info, too. Some of this info might be available on the Info Centre pages... check out the top of this page and look for the drop-down menu options... General, Hosted Sites, Civ 5, etc... hover your mouse over top of the Civ 4 one, and then you should get links to an Info Centre page for each of BtS, Warlords, and Vanilla (although they don't use the wording "Vanilla").
Good luck!