No 'horrible entrance' at all! Welcome to the forum, Ralf and Dido!
Now, the diet is not very good. Nothing wrong with feeding budgies seeds as long as they are the right kind but, in my personal opinion and experience, it's much easier to feed them gloop and greens than to go through all the gyrations of trying to convince them to eat healthy stuff along with the seeds on their own. Right now, I only have seven budgies, four females and three males, so I do need another male to make 'things' even
but I've had one of my sons and his family staying over for a visit and I had no time to look around for one that might need a good home [I never buy baby birds, I take in the ones that other people don't want and, with budgies, it's an easy thing to do because people are always buying some and then giving them away]. I feed mine gloop with a piece of fruit or veggie [it depends on what they like best of the ones I have for the day] for breakfast and all day picking and ABBA 1600 for dinner [they also get ABBA's multivitamin/mineral supplement twice a week and have cuttlebone in their cage all the time]. Gloop is a dish made out of cooked whole grains mixed with chopped veggies and flax seeds [I also add sesame seeds right before and during molt]. For example, yesterday, they had blue curly kale and cucumber [they actually love the kukes if you make a thick slice of them and skewer them on the tip of branch] and cinammon flavored gloop. Budgies are ground foragers so putting their food on the bottom of the cage on a white paper plate helps a lot because it's the way they would eat in nature. And they LOVE LOVE LOVE leafy greens! Try putting the very heart of a romaine up high in her cage and make sure it's dripping water [they love to take a bath by rubbing their bodies on the wet leaves] and you will see that in a day or two, she will start eating it in earnest. BUT for them to eat a good diet, you can't free-feed them protein food because, if you do, all they will eat is the protein food and not anywhere near enough of the good stuff. What I do is give them the greens and the fruit/veggie first, wait about half an hour and then give them the gloop.
As to the 'sleep' schedule... well, the thing is that it's not how many hours they sleep but how many hours they are exposed to light. Budgies are extremely opportunistic breeders [so much so that you will still find references in birdsites stating that they breed all year round] and, because we always feed them too well and always keep them in a steady good weather environment indoors, if you don't follow the solar schedule strictly, you end up with birds that are hormonal all year round -which is extremely unhealthy for them. Now, the key to a solar schedule is the exposure to dawn and dusk without any artificial lights on because it's the different spectrum that happens at twilight that turns on or off their 'internal clock' which, in turn, regulates their endocrine system.
As to training... well, you are not going to like what I am going to say but please understand that it's nothing personal against you, it's that I care very deeply for birds and want them all to have the best possible life which, for a budgie, means living in a flock or, at the very least, having a mate. Lone budgies are unhappy little birds. Lots of people say "My budgie loves me and it's not unhappy" but it's wishful thinking because it's a known fact that undomesticated animals are not happy when they live a life completely different from what nature evolved them to have. So I would beg you to consider this.
The other thing is that although you can train a budgie, what you can get from the bird is actually very limited because they are naturally flighty - and that's why she will not stay on your finger. Budgies have 'ants in their pants', they are always moving, flying, hopping, jumping, hanging from something, exploring a little cave, chewing on something, preening each other, etc. They move too much, their attention span is short and they are not, unlike companion parrots [budgies are aviary, not companion], fixated on one single individual [they are flock-oriented and not pair-oriented and, to put the icing on the cake, they are hardly ever hand-fed] so, although you can teach them to come to you, to perch on a finger, to take a treat from your hand, etc.
you can't really expect them to stay still long enough for them to learn anything else because you would be working against their very nature.
If you want a single bird that can be trained and would bond deeply with you, get a GCC BUT ONLY if you have the hours and hours they need of one-on-one during the day [can't do it at night, it screws up the bird endocrine system] because they are extremely needy little birds and need A LOT of personal attention [much more than other species]. And do consider adopting instead of buying. For one thing, we have a huge overpopulation problem with parrots and you wouild be part of the solution instead of contributing to the problem and, for another, the notion that one needs to get a baby bird to get it to bond is not true, all my birds came from somewhere else and they are all wonderful animals.