I don't mean to rain on your parade, but you'll not likely have much success rooting cuttings of oak. It has been done, but not with high % success. Working with 'sucker' material will increase your likelihood, since it's 'juvenile' material, but I still wouldn't get my hopes up much. In general, with most hardwood cuttings, shade - or at least filtered sunlight, with protection against brutal afternoon sun - is helpful, and creation of a humidity chamber(can be as simple as a clear glass jar inverted over the cutting) can be helpful.

Most rooting hormone products contain anti-fungal ingredients. May help shouldn't hurt, but not necessary for rooted plants.

Rooting hormone is butyric acid ( labeled a bioactive chemical ) and as a cutting stimulate is ideal. But this acid can damage mature roots or even plants nearby. It is actually considered a hazardous material and you are cautioned not to place it near water supplies. Might just me a "don't sue us if your child gets sick on it' warning but I wouldn't use it on a mature plant. Alfalfa meal is what many gardeners use to help the roots mature. Alfalfa has a hormone that stimulates growth.

Sure it is. Many millions of plants were propagated from cuttings before rooting hormones came into existence. The three keys to success are: proper timing in taking the cuttings, a suitable rooting medium, and a means of maintaining 100% humidity around the cuttings until rooting has taken place. Old time propagators used sand, mixtures of peat and sand, or anything else that held moisture but had lots of air space. Containers with good drainage, a glass cover- later a poly plastic one - to maintain humidity and a lifetime's experience to know when to take the cuttings.