1) What is "celebration"? What are its causes and consequences? It seems to be caused by happiness and lead to increase in population. But more in detail...

2) What is "martial law"? Is it about some military units making city riots less likely?

3) Sometimes (!) I can't research Religion from the start! It is generally so, that sometimes some advances are not available for research in the science dialoge, although according to the great library, it should be. Often, an advance is available in one turn, but if I choose to research another advance, it isn't available again in the next turn! So I don't know what advance will be available when and therefor I can't plan my science as I want. It all seems so random.

Is this simply a bug?

1) Your cities celebrate you from time to time, when they are very happy. I think happiness>95% causes celebrations. They are very good, because the give bonuses in your civ count.

2) Theocracy e.g. makes martial law. Your citizens are happier with it. Before change of government you must count in that you will lose this bonus.

3) You start with some advances already known. Perhaps you already know religion so you do not need to learn it.

I think the random advance availability thing is unfair in multiplayer games. It makes chance, not the players, decide who will get what early wonder. But if it has to do with deity, I'll just play multiplayer with a lower difficulty level (never tried that). (Unless there is a file to edit in order to remove that detail from deity).

About celebration:
How the heck do you reach 95+% happiness?
Maybe I'm too cheap towards my subjects, maximising prod and food while keeping wages low...

Hm, the martial law should perhaps someone explain who knows the .txt files exactly. I can just explain what I noticed during the game.
Concerning the happiness: Further in the game you can build more City improvements that make people happy. The wonders like Contraception increase it. And when the population is large enough you'll get many entertainers.
So I get easily 95+%. But I played it only in chieftain level, higher levels are more complicated.

Celebration occurs earlier than happiness 95. I can't remember the exact figure right now... maybe 85? It's useless, so I never cared. And it clutters up your screen with those "!" icons....

2) Military units in your cities exert martial law under many government forms. E.g., under Fascism, you get 2 happiness points for each military unit (up to 2 units, IIRC) in each city. The effect isn't very big, but it's helpful.

When Cities have a happiness of 85% or above they celebrate.

At the higher levels of dificulty this meens that the cities grow twise as fast as they would otherwise do. That is: The amount of stored food needed to grow is reduced by 25% if you have graneries (not sure about cities without)

Apart from that Gregurabi is right.