2) The play compels us to choose between those who are ruled by the head, and those who ruled by heart. Do you agree?
Initially you need to explore the terms ruled by the head and ruled by the heart and distinguish between the two, and indeed the characters that fit into each category. It could be argued that there is not a simple dichotomy even though there seems to be one superficially.
To support the concept that the characters are divisible into two categories we can use the argument that the characters are inevitably polarized in a play that is posited so strongly on war, conflict and relationships. Thus we are compelled to choose even if it is simply due to political or even comical motivations. On the other hand it could be argued that we are not forced to choose because the characters are not dichotomized in this way. There is not a simple division to be made between them. Yes, we do respond more positively to some characters than others but the reason for this is because they are mixed and not made up of defined, specific character traits.
Certainly there are more cerebral characters. Henry IV is perhaps the most obvious of these yet even he has a highly emotional response to Hal and the troubles of his country. He is disappointed in his son and outraged by the impertinence of Hotspur both very emotional responses. Furthermore we only ever see him as a King, never in a personal / human capacity (except perhaps in Act III, Scene I) thus it would not necessarily be fitting if he were a highly volatile, exuberant character. Hal too could be seen as more cerebral, certainly calculating even manipulative, yet questions can be raised as to the nature of his relationship with Falstaff. In this he shows emotion and the language of his soliloquy and of the repudiation of Falstaff need to be considered.
Whereas the cerebral characters could be considered as admirable and intelligent, the emotional characters are equally as compelling. Falstaff essentially could be categorized in this way through his extreme joie de vivre and pure affection for Hal. Yet at the same time there is some indication that he is also thinking and calculating clearly ("When thou art king"). Hotspur is also a highly passionate character yet we need to consider what he is passionate about his wife, or his military prowess? In his language, actions and attitude there is evidence that it is both - that he is both cerebral and emotional.
It is not a play that asks its audience to make such clear-cut decisions. We are not presented with such one-dimensional, uncomplicated characters so we are not forced to choose.
2) Would you agree that Henry IV, Part I often gives no clear view of how a state is best governed?
Certainly within the play we are presented with many models for a leader, however, few of them seen wholly appropriate. Even the future monarch (Hal) does not seem to be an ideal. Is it significant that we never see him as the King in this play? It certainly needs to be questioned whether or not Shakespeare needs to present Hal as an ideal potential monarch for propaganda reasons for either the audience or the Tudor monarch.
As this is a play of issues and ideals obviously a number of questions are raised thus are we forced into a conclusion of how a state is best governed? The rebels are crushed therefore their ideas are proved to be unworkable (as they are obviously shown to be throughout the play). The play does move gradually towards order Henry IV triumphs ad Hotspurs death are both very symbolic one set of ideals triumphs over another.
Henry IV is a detached monarch, out of the public eye and elusive. He is the central form of power within the play an absolute monarch and part of the traditional hierarchy. Nonetheless he is haunted by the past and is thus highly aware of the need for a stable succession. Richard II had much more contact with the common people as a monarch (this is certainly Henry IVs opinion). He is derisive of Richard IIs need for popularity and sees this as a justification for his deposition this is the link between his son and Richard II that worries him.