Okay. I will ask 3 questions.
1. When should a person do Ijtihad?
This basically answers that:
It was narrated from Shuraih that:
He wrote to 'Umar, to ask him (a question), and 'Umar wrote back to him telling him: "Judge according to what is in the Book of Allah. If it is not (mentioned) in the Book of Allah, then (judge) according to the Sunnah of the Messenger of Allah [SAW]. If it is not (mentioned) in the Book of Allah or the Sunnah of the Messenger of Allah [SAW], then pass judgment according to the way the righteous passed judgment. If it is not (mentioned) in the Book of Allah, or the Sunnah of the Messenger of Allah [SAW], and the righteous did not pass judgment concerning it, then if you wish, go ahead (and try to work it out by yourself) or if you wish, leave it. And I think that leaving it is better for you. And peace be upon you."
2. Which person can be considered Mujtahid besides Hakam?
Basically anyone who has sufficient knowledge in various branches and can perform ijtihad independently
3. Can a person put a condition in giving bayah to the Ruler? If yes, then is that condition applicable to only one Ruler or all Rulers? i.e. Muawiya put forward the condition of Qisas of Uthman when asked to give bayah to Imam Ali (a.s) but when Muawiya himself became ruler no such condition was put forward by people when asked to give bayah to Muawiya.
As a decision maker you mean? Basically yes, there is no explicit order that forbids you from doing that. However the condition can not be contradictory to the Qur'an and Sunnah and there must be a great interest served by that, that supersedes the interest served by not placing a condition and giving bayah directly if other decision makers have already preceded in given bayah without such a condition. This is what Sunni's later on realized, agreed upon and canonized more or less that such interests rarely exist. But again, this was later on.
As for another ruler, it is at the discretion of the decision makers if you again want to put that condition, so if they feel that such interest does not apply anymore or does not supersede other interests, they may forego placing such a condition.
People gave bayah to Muawiya without demanding Qisas of Uthman whereas in the case of Imam Ali (a.s) people demanded Qisas of Uthman before giving bayah. Why this inconsistency?
It is not inconsistent if they viewed placing such a condition for Muawiya (at a later point in time, don't forget) not serving any interest anymore or not serving it enough to supersede the interests of giving bayah directly.
I have a feeling that you will not be satisfied with such answers until someone gives a detailed explanation of what was going on. The problem is, we can only speculate. One problem during the time that I think you are overlooking and could be one factor is the following:
First of all, the murderer of Uthman was not known specifically. However, many people were responsible for it and they joined Ali's army. One factor in Muawiya's reasoning could be that they should be punished, not necessarily through their death as they were not murderers per se, but not letting them benefit from their actions. One of the greatest benefits of the death of Uthman was their gaining of power and influence under Ali. If you want to couple this to an Islamic principle, Sunni's believe that a murderer can't inherit from the murdered. So Muawiya was basically telling Ali to get rid of these rebels and take away their power and influence. Ali by his own admission could not do this because they were too strong and would led to civil war. This was unacceptable for Muawiya because this basically meant that Ali could not assert his authority. He was basically telling him, how can you demand obedience from me if the very people who pledged allegiance are not even obedient to you. If you can't assert authority then you are not fit to rule in the first place. It's actually one of the conditions of Sunni's as well that a ruler must be sane, must be physically well and must not be a captive. Muawiya was basically telling him that he is held hostage by his own army and therefore as long as that is the case, obedience to him is not obligatory.
And if you look at the popular account of how the Battle of Siffin unfolded, you can see how disobedient they were. The armies of Ali and Muawiya camped at Siffin and Ali was still negotiating. His army attacked and Muawiya fought back. Ali didn't gave orders to do that but saw himself forced to start the battle. When the armies called for arbitration, Ali ordered them to continue, they didn't listen and wanted arbitration. Ali saw himself forced to accept arbitration. Ali wanted to send Abdullah ibn Abbas as arbitrator or someone else, they wanted Abu Musa al Ashari. Ali saw himself forced to listen. Abu Musa al Ashari declared Ali to step down, he couldn't accept. Now his army declared him a kafir for allowing arbitration in the first place! Ali saw himself forced to attack them in Nahrawan. This is exactly what Muawiya didn't want in Shaam, because they were inherently rebellious and anarchistic and their power had to be taken away.
To come back to your question, this also explains why such a condition was not necessary for Muawiya, because they were never part of his army and had influence or power over him, not to mention that a lot of them died in the battles of al Jamal, Siffin and Nahrawan.