do cyclones/hurricanes/typhoons ever cross the equator?

Tropical cyclones/typhoons/hurricans cannot form within 5° of the equator. They need Coriolis to get them revolving. Coriolis is proportional to the sine of the latitude. The sine of 0° is zero so there is no Coriolis at the equator. The systems must form further away. Once they have formed, the winds take them further from the equator usually in an easterly direction at first and then, as they get further from the equator and into the westerly wind belts, the move to the west. As they must form at least 5° from the equator and there are no winds to take them towards the equator, they can never cross the equator.

No;they usually move away from the equator;they curve towards right in the northern hemisphere and towards left in the southern hemisphere due to coriolis force.Coriolis force only gives the spinning motion to hurricanes and this force is absent over the equator.Hence if any hurricane comes near the equator it may get weakened.

Tropical systems will always form just to the south or just to the north of the equator then move away from it. They cannot cross the equator because the winds will be opposite from the areas just to the north and south. The change in prevailing wind direction would destroy the storm.

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No they do not. They initially can move somewhat parallel to the equator, about 5-10 degrees either side of it, but eventually do turn poleward with time.

No, that's never happened and might never happen. They always form either north or south of the equator.