No. The excellent circulation to the rectum and the resulting delivery of infection fighting blood cells make hemorrhoid infections pretty rare. The itching and redness at the rectal opening is often complicated by yeast (monilial) colonization from the bowel. I've found the use of an antifungal (eg Clotrimazole which is over the counter) and some cortisone cream very helpful to get rid of the itch.
Not typically. Hemorrhoids do not tend to get infected, but they can, especially if they develop a clot.
Unlikely. Hemorrhoids only very rarely become infected, despite the presence of stool next to an open wound. This is a testament to the efficiency of the immune system in the rectum, specifically the gut-associated lymphoid tissue (galt). The more likely scenario of infection is the case of hemorrhoid strangulation, where the infection sets in when tissue dies. Thankfully, this is rare.