I have read your essay and it certainly contains some points of merit. I do, however, think it is rather biased towards the Queen model.
First, some factual points:
You talk about religion. The British prime minister is not allowed to be a Catholic. Therefore, perhaps you should ask whether the same might apply to a British President.
You should refer to 'laws' rather than 'legislations'.
Try to get the language right. A General Election takes place when the Queen accedes to a prime minister's request to dissolve parliament. Similarly, it is by convention that after an Election the Queen asks the leader of the party with the largest number of Commons seats to become prime minister and form a government.
Now, some omissions from your essay:
The Queen is a constitutional monarch. What this means is that the royal prerogative powers are exercised on her behalf by the prime minister and his Cabinet colleagues. Government is conducted in the Queen's name. In other words the Queen 'reigns but does not rule'. In principle she could dismiss a government at any time, though this has not happened since George III dismissed the Fox-North government in the 18th century.
It is highly likely that a president would be elected - as in the USA. This in turn suggests that there would need to be a written constitution, whereas at present the UK has an uncodified constitution. Such a written constitution is also likely, as in the USA, to embrace the principle of the separation of powers. Hence the president would be head of the executive branch, but would not be responsible for law-making (the legislative branch) or the upholding of the law (the judicial branch). The only alternative to this would be a despotic president with supreme powers, but this is clearly not an option for the UK.
So far as the armed forces are concerned, the president is likely to be commander-in-chief, as in the USA. This means that he is ultimately responsible for deploying troops, though you should note that in the USA it is Congress that has the power to declare wars, ratify treaties and finance military operations.In the UK, it is the prime minister who at present makes all these decisions, subject to parliamentary approval.
Note that if the UK followed the US Constitution, there would be no question of the president granting honours, as you suggest, since the Constitution explicitly bars the granting of honours and titles.
The key point that needs to be made about a president is that he would almost certainly be elected, in contrast to the Queen, who is an hereditary monarch. An elected president would therefore have his own mandate, and would automatically be the spokesman for the UK. He could also be defeated in a subsequent presidential election by another candidate. He would combine the roles of Head of Government and Head of State. This is unlike the present situation in the UK, where Gordon Brown is Head of Government and the Queen is Head of State.
Finally, the interesting question to ask is whether we could have both, as in some European countries. The Queen would then have a purely symbolic role, and almost certainly no constitutional role. We would no longer talk about Her Majesty's Government, courts, etc.
I hope this is helpful.