They don’t have to be peeled. The more finely you chop them, the better they will extract into the alcohol. If you want to make it stronger, you can fill 1/2 of the jar with roots, and then add the alcohol. You could also try making a 1:1 tincture meaning that for each gram of root, you add a mL of alcohol.
Keep in mind, the roots are best harvested in the early spring or autumn when the plant doesn’t have a lot of other activity like growing leaves and flowers. The older the plant, the larger the roots will be. First and second-year plants are going to have quite small roots. Rather than a full glycerine extract, you could add some glycerine to the tincture to add additional sweetness. Glycerine is not nearly as strong as a menstruum as alcohol. For the root, alcohol or water by means of decoction, or capsule form is ideal. Consider, what benefits are you seeking from the dandelion root? Remember that if you’re looking to support the liver, alcohol can contradict this and capsules or decoction can be a better choice. However, if you’re looking for a gentle bitter or a laxative effect, the tincture can work well.
A tincture can certainly be made with dandelion leaves! Though often, a dandelion leaf is used for its diuretic properties and for this, an infusion is a preferred preparation. I think ginger, mint, and dandelion leaf tea sounds delicious. You could add something like marshmallow leaf or chamomile, or serve it with some honey to moisten and smooth it out.
The flower heads are primarily used as food, but I have also heard of them used in oil for skin support or as edible oil. This is a better use for them than tincturing. You can also use the dried flowers in tea!