Is this the correct way to handle the situation?
No. But neither, rather contradictory, is this: "You are allowed to make a fool of yourself as long as you show you can and will grow."
The reason why is simple - no-one is perfect, as you've said. That means no-one has a right to declare someone else as being a fool, nor does anyone have the right to dictate how people should develop as individuals. As immature or distasteful I or anyone may find something, I don't think anyone is entitled to request that person changes their view [grows] to conform to someone elses view. The best response is to simply state we are all human - but also all individual. There will be a reason(s) why people believe what they believe and argue their points, and those reason(s) are just as valid as anyone elses, regardless of our stance on them.
'can a couple tweets/statements/actions really point towards a systematic hatred of a race/group/person/idea?'
Not at all. I find [personal - being gay] that any homophobic person who meets gay people suddenly realizes we aren't evil or immoral, even if they find us distasteful. Twitter and the like dehumanizes us and puts us behind a veil - even the most bigoted person can come around when they realize the real-world effects and suffering their view could cause.
'can a couple tweets/statements/actions really negate actions you made in the past?' It is impossible to negate an action, but there also isn't some morality scale that people can be judged on based on past or present actions. As a general rule assume in good faith that whatever the individual is saying at that moment is their genuine feelings for that moment, and seek to understand why and/or how their opinion has either stayed the same or changed. As for past actions, I simply make myself aware of them. There's a difference between good-natured interaction and blind idiocy - being aware of what someone has done gives you insight into what they may still be capable of.
'if it is a measure of time, how long is acceptable?' There is no arbitrary measure. But there's a fundamental distinction here - between words and actions. If someone says something, they can retract it or reject it. If so, I will immediately act on their new beliefs in good faith, unless there is a good reason to suspect they aren't being genuine. They don't need to apologize for what they used to believe or what they presently believe, but I generally prefer a reasonable explanation of the shift in thought.
Actions are different - they have actual effects in the real-world. For actions, the only forgiveness a person should seek is from the people they may have wronged, if they wronged them and if the person needs that closure. It is entirely between the two people, and isn't the business of any bystander to judge.