Many times; in multiple versions of the Robin Hood tale, there are instances where characters mention that death is the penalty for taking one of the king's deer. In that vein, my question is threefold.
1: Was there a time period where a man would hang for poaching in England.
2: Were lords partially or completely exempt from this penalty, as members of the nobility.
3: What were peasants allowed to hunt on their lord's lands, if deer belonged to the king.
I appreciate any answers to this, and thank you for your time.
Yes, No, In theory anything not explicitly protected. In practice bans on anything that could be used for hunting would have made it tricky.
Due to the way the laws worked 1 is a completely different question to the other two (you would need someone who knows their way around 19th century laws to find when people stopped being hung for poaching or related activities).
So lets look at the other two. If something is a king's deer that means its in a Royal Forest (feel those capital letters). Robin Hood is set during the reign of King John when Royal Forests were pretty expansive. In fact expansion of the forests was one of the issues raised on the Magna Carta. Having an area declared royal forest was pretty financially devastating since you couldn't really farm it it (you might be able to run animals on it in much the same way you could on a common) and without the king's permission you couldn't hunt in it. If you were a lord you might be able to buy the right to hunt in a forest but such rights could also be given as a reward. In 1361 Sir John de Sully was awarded “Once in every year during his life, in any of the royal forests, parks or chases in the realm, to have one shot with his bow, one course with his hounds, and one chase for his dog called “Bercellette””. Sir John would have been about 80 at this point so I'm not sure how much he would have used the right. So for question 2 no a lord didn't have a right to hunt in a royal forest but they could probably buy the right.
King John's expansion of the forests was a a pretty big deal at the time Robin Hood was set. In fairness he wasn't the first two do this. King Rufus (William II) did similar things but had the advantage of suffering a sudden case of accidental arrow to the chest (in a royal forest) before things could really come a head.
To be clear just because land was on in a royal forest doesn't mean it belonged to the king. It just meant if you did own any land within the forest you were rather limited as what you could do with it. You see how expansion causes issues. So for question 3 that the land belonged to the peasant's lord wouldn't be the legal issue (what do you mean by peasant here? Villeins only or are you including freemen?)
With King John's death the job of gluing the kingdom back together fell to Henry III's regent William Marshal. As well as issuing a a slightly modified version of the Magna Carta he also issued the Charter of the Forest which reduced the size of the royal forests (nominally to the size they were during the reign of Henry II) and reduced the penalty for poaching to a large fine. They did however continue the “lawing” of dogs (cutting of part of the front toes to prevent them being used for hunting). You can see the original Charter of the Forest yourself at Lincoln castle but its in abbreviated medieval latin. Might be better with this translation:
http://www.nationalarchives.gov.uk/education/resources/magna-carta/charter-forest-1225-westminster/
Over the centuries royal forests generally declined as and the crown's control over them declined with the power of the crown. By the time we hit the 20 year rule there are only a handful left and the legal oddities mostly relate to rights of common rather than any royal claims.