A:
Fortuitous and lucky are very similar in meaning, often used interchangeably. 'Fortuitous' can be a bit more literary sometimes, a bit more serious or formal sometimes, a little old fashioned sometimes. "Lucky" is less formal, more casual, more likely to be said in conversation. In meaning, they are mostly the same.
The main difference is in nuance and context: 'fortuitous' is a little bit more like 'this was blessed,/ this was favored to happen this way / I was fortunate because of this happening.' Almost like someone is looking out for you. It was fortuitous to run into someone that you've been meaning to talk to so now you can talk them. Almost like someone planned it out for you or was helping you achieve it. It was fortuitous that the item you need to buy was on sale. Almost like fate was giving you a little help with this and approved of your plans.
Lucky is a bit less special, a bit less like someone is personally looking out for you and a bit more like random chance was momentarily in your favor. You got lucky when the History Test was rescheduled so now you have more time to study. You found some money on the ground, so you were lucky. That could have been dangerous to do, but you were okay. So you were lucky that time but don't try it again! Lucky feels more random, more temporary. Some people are lucky a lot, but people always expect their luck will give out on them sooner or later. No one is lucky forever. Luck cannot be counted on.
Coincidence is different from both in meaning, but it can sometimes happen at the same time as luck or fortuitous events. Coincidence just means that two things that seem to go well together, happened almost like they were coordinated but no one actually planned it, it just happened that way. It's like plans that no one purposely made. So if I happened to see someone when I got on the elevator, and I had just been thinking about calling that same person on the phone later, that is coincidence to run into them while thinking about them. It's fortuitous to have the chance to talk to them just fall into their lap, as if fortune favored your plans to talk to them and helped your plans along. It's lucky to see them there, because it saves you a step. It's coincidence because you were just thinking about them and then happen to see them in person, almost like it was planned even though it wasn't. All three words work for that situation, but all for slightly different reasons.
But coincidence isn't always lucky. Maybe you had a bad feeling when you woke up today and coincidentally you receive bad news several times that day. It was probably just coincidence, but it feels like the timing of the bad mood and then bad news all day was pre-planned somehow. Oddly coordinated. Arranged by chance to happen together even though no one planned it. Maybe it's not bad luck or good luck, it's just odd. Like you noticed one co-worker was getting to work at the same time as you did today, and when you leave work that day you notice they happen to be leaving at the same time as you, again. Even though normally you don't see them because they work in a different part of the building. So coincidentally you both arrived and both left at the same time, almost like it had been planned even though it wasn't. That was coincidental. Or maybe you turned on the tv and heard the same commercial that was playing on the radio when you turned off your car, which should have been random but instead it was almost coordinated together, which is kind of weird or funny. That's a coincidence. But not really lucky or fortunate that time. Just coincidental. It might not mean anything, but it's a strange pattern, almost like two unrelated events were meant to happen together. It might not be helpful, but it's still a coincidence.