I hope that I can help by adding the following thoughts and advice from an actual denture wearer:
(i) you should probably discuss all of your questions and concerns with your dentist (as opinions and practices vary from dentist to dentist);
(ii) I agree that you should be cleaning your denture at least daily, but, in my experience, what you should be cleaning it with depends upon what kind of liner (if any) is in your denture (because commercial tablets can damage soft liners) as well as your personal preferences (in my case, I find that Joy dish wash detergent is both gentle and effective for most cleanings with only periodic soakings being needed) and I have been advised by my dentist to use ultra soft (or baby) toothbrushes instead of denture brushes because they are softer and less likely to damage acrylic teeth;
(iii) I was advised that, especially at the beginning when they are healing, it is not advisable to brush gums but, instead, to rinse entire mouth and, if necessary, swab gums with wet gauze (because you don't want to disrupt healing or fit or your denture)'
(iv) your goal should be to get your denture to the point where you don't need adhesive and, in this regard, given the shrinkage of gums following extractions, one or more soft relines can be extremely helpful (and, in fact, enable me to be free of adhesive, including that horrible feeling when it starts to dry out and the awful clean up after using it);
(v) yes, there are different types of soft liners and some work better than others, but, in general, they are really helpful and there is no need (assuming you are willing and able to pay the cost) to wait a long time to get your first soft liner (and, in this regard, I had mine about a week after my extractions);
(vi) from what I have learned, it is really not possible to have the palate part of an upper denture removed and maintain the stability of that denture in your mouth; but it should be possible to have your denture fit so that, after some practice, your speech is just fine; and
(vii) once your swelling has fully subsided, if necessary, your dentist (or his or her dental lab) should be able to thin out the part of your denture that supports your upper lip.
Good luck. Hang in there. It will get better with time, patience and practice.