It is unclear whether the contest involved shooting the arrow through the rings attaching the axe heads to their handles, or through the rings at the end of the handles designed for hanging them as ornaments onto the wall. Since the former would require the archer to lie on the floor, the latter seems more likely.
In order to loose an arrow, the string that connects the tips of a bow must be stretched taut. This means that when stringing the bow, its limbs need to be bent so that the string can be slipped over. Depending on the strength of the bow, this can be an extremely difficult and physically exerting task. Penelope’s task is therefore twofold; a test of strength must be passed before even attempting the test of skill.
Tallow is rendered animal fat, like lard. Warming and greasing the bow would aid the process of trying to string it.
The pollution being referred to here is not literal, but religious. The association of death, especially violent death, murder or suicide, was thought to pollute the area where it had taken place, as well as any objects involved (such as a noose). Pollution repelled the gods, which is why nothing associated with death (or birth) was allowed in sanctuaries. To expel the pollution, the area needed to be ritually purified; this might be achieved through fire (burning the pollution away), or with pure water (such as washing hands before a libation or sacrifice to the gods, or before entering a sanctuary).