It could depend on the flute and the octave. In my flute I cannot hear any difference in the first octave.

But a trill on G sharp, which should be done with two LH fingers also sounds identical if trill with just one finger, but it is not the case with my teacher's flute.

I trill this way because it is much easier for me and I get a clearer sound, but it is not a good idea to use alternative fingerings as a student, if there is no real need for it.

And yes, I find the instrument more stable with the RH pinkie down.

Normally the D# key is left open to allow for the expected venting. A note is determined by the first open hole the air column reaches (at least in the first couple octaves...after that, double venting starts to happen more often), and theoretically, anything below that hole should remain open to allow correct venting. On some flutes, and particular notes (E for example is more noticeably changed by allowing the D# to close than an A is), the change can be more notoceable than on others (as Jose Luis noted). You may find that it does not matter with your setup, but you should learn to do things the conventional way (i.e. with the D# key as appropriate) before you start departing from the standard way of doing things. In very quick passages where the pinky would be working pretty hard (such as a fast D-E-D figure), it may be alright to leave the pinky up, but each instance is different, and you have to decide when it's appropriate based on the music surrounding such a figure. I would encourage you to start breaking this habit....You may find that there is a difference you never noticed before, whether it be changing the sound of the notes slightly, or changing the balance to a more comfotable position, or something else entirely.

Most teachers will advocate that the Eb key is kept down for most notes as per a standard fingering chart. There are obviously some notes when this is absolutely required (Eb for instance) and others in the top octave when depending on the flute use of the Eb key (or other alternative fingerings) improves the tone and the ease of playing. With some flutes you need the Eb key down to play E readily in tune.

I tend not to use the Eb key (nor teach its use with young students) in the lower two octaves. This gets me into trouble with other teachers! As noted above For many players, holding the Eb key down does help holding the flute.

if you don't depress the Eb key, you will be out of tune in most octaves, sometimes on high E one can leave the Eb key up, but not on the lower octave, the notes would be terribly flat, I won't tell you what I tell students who have been given wrong fingerings by other teachers as I am not rude on this forum....

nor teach its use with young students

In my opinion, that's doing them a rather large disservice. As has been noted in this thread already, sometimes you can get away without using it, and sometimes you can't, but IMHO, they need to start out learning the standard fingerings before they depart from them. They are standard for the reason that the provide the most well in-tune note with the best tone, and young students are not likely to experiment with fingerings or question what they've been taught. Perhaps you introduce the D# key to older students, but that's really just adding another layer of complexity to all the other things they're trying to learn....By that point they should have a good grasp of standard fingerings. You've already put in the time learning to play, so if you decide to depart from the normal fingerings, that's your decision, and does not impact anyone else, but to pass along that habit to your students is doing them a disservice, and presenting them with something they'll most likely need to correct down the line.

It will take some time to get there...It's hard to manipulate just your pinky, and I imagine the fact that you haven't used it consistently adds to that. Try slowing down the fast stuff to the point that you can play it well, and then gradually speed it up from there. With practice, you'll be able to use it as you should. Good luck....It's never easy breaking bad habits (especially if they've been with you for a while), but most times it's well worth it.

This thread reminds me of an occasion 30 years ago when a well known British conductor was rehearsing a regional youth orchestra and the flutes were out of tune with each other. Rather than advising them to listen either to each other or lower harmony (often more effective), he insisted they all put the Eb key down. Needless to say, the intonation was still faulty because the issue was that the flautists weren't making the necessary adjustments to accomodate imperfection in the scale of their instruments, volume dependency, harmony etc.

It is interesting to speculate whether in another 30 years it will still be fashionable to keep the Eb key down, whether there will be open Eb key flutes, or more recognition that holding the Eb key down isn't really necessary for good intonation on most notes in the lower 2 octaves(E and Eb excepted). One could also speculate whether changes in flute teaching have a significant difference in the number of students who give up. Some of the changes over the last 30 years are flawed - like not using 3RH for the standard top F fingering. If I'm around in 30 years time, I'll probably be past caring!