Let me try to touch on each of your points a bit.
It's always hard to say how transition on the job will go. As I'm sure you're well aware, in the worst cases it can result in harassment and/or getting fired, although if you say a coworker came out to him and things have been fine then it's a good sign. Honestly, if you are otherwise ready, you probably go for it at work, assuming that you could get by should things turn south. If it's not going to work out, better to find out sooner rather than later so you can find a job where it won't be a problem. Transitioning at work is scary, but there are certainly jobs where you can do it. My transition went about as smoothly as could be asked for and while that may be too much to ask for in most cases, it does show it's possible.
That being said, if you're not ready for transitioning at work, you may want to hold off with saying anything there for now. You don't want to come to them on a Friday and say "Hey, I'm going to be coming in wearing a dress on Monday, k?", but you also don't really need to tell them that you're going to transition in 6 months or a year, either. Don't give them too much time to get anxious about what that might entail.
Now, for the tougher part. The situation with your dad sounds really tough and I won't pretend to know what the right choice is. You say he's considered terminally ill and you worry about what your coming out to him might do to his already bad health. I suppose there are a couple things to factor in here. One, do you think telling him about your situation would make whatever time he has left more stressful or do you think it would allow him to see that you know what you want in at least part of your life and are working to make yourself happy? I think how you answer that really might influence your choice. And while this is kind of grim to talk about, when he finally does pass do you think you'll be OK going through the grieving process seen by others as a son? Will that feel like yet another lie to stick with you the rest of your life? Or will you feel it's the best/only option to give him a last bit of respect? There's not an easy answer there.
As for your cousin, well, I feel that's a bit easier. They will have to find out eventually, won't they? You can't hide it forever. Is it the best thing to bring to them? Maybe not, but that probably won't change if you wait either. It's great that you're showing concern for them, but you need to consider both yourself and what the best way for them to hear about your situation is. Better from you than from someone else, right?
Overall, just hang in there. I know you'll get through this.