First of all, ALL of this is dependent on your local regulations. In other words, you need to check the laws in your area.
The euthanasia drugs used by vets are NOT simply an overdose of sleeping pills. Euthanasia drugs work almost instantaneously and must be given IV or will cause undesirable problems (don’t remember exactly what). Obviously, your vet may be unwilling to allow an unexperienced person (namely YOU) to administer such a drug. However, you could certainly ASK the vet if it would be possible. Any good vet will certainly allow you to remain with and probably hold your pet while she is euthanized (assuming, of course, that you’re not hysterical or some such), even if they won’t let you actually inject the drug due to the technical requirements. I don’t believe any vet will give you “Euthanol” to take home; it is a dangerous substance and the vet is responsible for it. I would hope that you have a vet that you see regularly if you have 3 dogs, which would mean that your pet is NOT dealing with a stranger. However, if you live rurally, a farm vet (e.g. large animal) is much more likely to make house calls; your regular city vet probably won’t.
On the other hand, it is more than likely not illegal for you to kill your own dog, as long as you do it humanely. <No, I am NOT interested in ANY discussion as the humaneness/morality of euthanizing pets. I’m talking LEGAL matters here.> Again, please check your local laws to be sure - animal cruelty laws are local mandates. However, sleeping pills are notoriously tricky to administer correctly for the purposes of euthanasia, and the reference materials you are using are written for HUMANS, not dogs. Again, I would suggest talking to your vet. Explain what you want and why and discuss options and possibilities. Most vets are very understanding people and I don’t think that your vet will be under any obligation to report you to authorities for merely asking about these things, especially if they are “hypothetical” questions. Best case, s/he will actively work with you to meet your requirements; worst case, s/he will tell you it’s not possible to do what you want (which you can simply ignore, if you wish). Middle of the road, s/he may be able to help you with dosages & such.
BUT, burying your pet can definitely be illegal, depending on where you live. Check your city ordinances - most cities ban burying pets on private property for health reasons. If you’re rural, you may not have a problem. However, it’s not like they have a “dead pet patrol” combing the streets looking for offenders. I’ve known many people to quietly bury their pets in their backyards (not that I would do that; it is, after all, illegal and wrong!) with no problems. It’s generally one of those things that is only enforced upon complaints from neighbors, etc. As far as the vet goes, they will certainly offer/suggest that you allow them to handle the remains (generally by cremation), but you can simply tell them that you’ve made other arrangements. They can’t MAKE you leave your pet. (There are actually pet cemeteries & everything.)
When it comes down to it, I would say that your best bet is to talk honestly with your veterinarian, assuming you have one that you feel is trustworthy. (If you don’t have such a vet, then FIND ONE NOW.) They will most likely know the relevant laws (at least the ones around here do), and will be able to help you figure out the best way to handle your situation. You sound like your primary interest is the comfort of your dog, and that is something that a vet will understand and agree with. Now you need to be sure that you have the knowledge and/or materials to actually accomplish the task without causing your pet any needless suffering due to your lack of such.
In case you’re wondering, my spouse was an emergency veterinary technician for some years and I spent a fair amount of time in that clinic. Their main problem was people who REFUSED to sign the legal paperwork required for the vet to put down a pet because “I just couldn’t do that” to the pet, preferring to leave their animal at the vet’s to die a slow and painful death. I know their reaction to your situation would have been to give you as much help and/or information as they could.
The adage “Knowledge is Power” is incorrect. The correct formulation is “Knowledge that other people don’t have is Power”. - The Donald