Excellent. I honestly think that the problem may be occurring at the crossover point when the cursor enters the input field. The three seconds were just a guesstimate at how much time we should wait to be sure that the drop mechanism (KeePass2 to Firefox) gets itself sorted out before we then create another event, namely entering the input box.

I think you should concentrate on pausing for three seconds before you enter that input field - test it out some more with no variations. If you can get close to 100% success then try reducing the time interval to two seconds. With all the hassle of switching browsers and/or password retention program, you don't need to get 100%, if you can get close enough that there's only a very odd failure then you can stick with KeyPass2 and Firefox.

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Right!
I'll give that a go and report back. Might be a few hours, but don't worry.
Thanks again.

Firstly tried pausing three seconds outside login field.
1. Ten out of ten times successfully.
2. Repeated with different email address, same success rate.
3. Different email address, no pause, dropped straight into login field ten times successfully.
4. Different email address, no pause, dropped straight into login field only failed attempt number seven out of ten times.

Suddenly, KeePass2 wouldn’t let me drop into Firefox any more. I had a Thunderbird Lightning Calendar entry field showing in a window below KeePass2, I could drop into that alright.

Closed & restarted Firefox. I noticed that trying to drop immediately into the gmail login field, Firefox wouldn’t allow it (the mouse became a circle with a diagonal line through it) for about three seconds as I hovered over the field, after that letting me drop into the login field.
That “enforced pause” happened a few times.
Failed on attempts eight & ten, out of ten attempts.

It looks as you say, that pausing does reduce the instances of the Chinese characters. Although occasionally I can drop into KeePass2 without an error, but that’s just the way it sometimes happens.
When Firefox stopped accepting drag & drop, for me it was the first time that had ever happened, so I didn’t read too much into that.

I didn't worry about reducing the pause time to two seconds, as three seconds is pretty negligible anyway.

Hydrurga, what are your comments on the above? Any more testing suggestions?

One day, research papers will be written on this subject. ;-)

I would say that the strike rate you have by pausing before you enter the login window is good. Just stick to that. Use the method for a week or so and see if you can live with the occasional Chinese characters. If you can, good stuff. If you can't, you'll need to go for one of the alternative browser/password manager combinations that were discussed earlier.

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It does seem a lot better with the pause. I'll see how it goes. I'll also mark it as solved.
Thanks to you and everyone else for the incredible support on this issue.