My wife and I are working on a divorce. Is this something I need to talk to people at work about? Everyone at work know that I am married and frequently ask questions like what I did in the weekend etc. Should I lie or not talk about it at all? Is there any advantage or disadvantage of talking about this to my manager?
This is, unfortunately, completely and utterly dependent on your specific workplace and social norms, and how much of a personal relationship you already have established with the individual people. While it is true that generally one should keep personal and professional matters separate, and there is certainly such a thing as "inappropriate personal disclosure", in most workplaces there is no vow of secrecy that prevents you from talking about anything about your private life. So I'll stick to some general advice to help you navigate the situation.
For one thing, if you don't want to talk about it, don't. Politely change the subject, or only share details you are comfortable with. If you spent most of the weekend fighting with a spouse and meeting with lawyers and someone asked how your weekend goes, you can just tell about the parts you are comfortable with (like all the junk-food you ate, or how you are trying to catch up on your sleep).
If you'd like to talk about it, but are unsure if it's appropriate for that person, you can be circumspect and answer questions about how your weekend went with things like "oh, just going through some hard personal stuff, but I'll get through - how about you?" If they are interested and want to inquire further, they can ask - and if they don't want to know, they should gloss over it.
One warning: people at work who are willing to talk about personal things are not always...discrete. That is to say, there is no assumption that what you tell to one person won't be shared with anyone and everyone else. If you are ok with people being aware of your situation, then talking about it should be fine - but if you want to keep it among few people, I don't know that you can do that without telling no one.
When people go through hard times - marriage, divorce, new family members, loss, trouble with children, etc - it's often possible to find lots of support and encouragement through your work. Many people reach out to people who they know are having a hard time or change of life, and you can meet new friends, get invites to do things you previously weren't able to do (or assumed you wouldn't be interested in), etc. But every culture is different, and not everyone finds a supportive environment at their work, so you'll have to be somewhat cautious and use your own best judgement.
I've found it more often than not valuable to share at least some personal details, especially about major life events that will tend to effect you whether you want them to or not. If you find yourself withdrawn and down at work and you just aren't your usual upbeat self, people are likely to notice and having a good explanation rooted in reality is often better than just letting people guess.
Finally, it can be good to try to find someone to talk with in these difficult situations, which are trying no matter who you are. If you have access to a therapist or counsellor, through your own or through something like a "family assistance" or counselling benefit at your work, it can be good to try to take advantage of those, as they can provide some more specific guidance and outlet for things you are unsure about sharing directly in the work place. Some workplaces also have legal benefits that can help you find a lawyer (or even partially pay for one), which is also recommended in challenging legal situations like this.
Good luck in making it through the difficult period you are in, and I hope you find the future to be far brighter than the present.
Should I lie or not talk about it at all?
I'd say that lying is no option. At some point you will get into a situation where you have to come out with the truth.
I'd choose the defensive option: Be honest but short.
I think most talks on work about family are just small talk and most people will avoid to ask further on such topic.
It's up to you. I suggest that you either say nothing or say that you're getting separated.
The advantage of saying that you're getting separated is that the message is succinct and it's unlikely that the colleague that you're talking to will pester you with "how was the weekend with the spouse?" questions going forward, unless your colleague has short-term memory issues.
Treat your manager like any of your colleagues.
Depends a bit on culture and the characters involved but in general a good manager can be a real help in this situation. Chances are this will affect you emotionally, you may have to move, and you may have to take some time off for administrative stuff. This is likely to affect your work, so given your manager the heads up can he helpful.
I had to deal with this in my team on multiple occasions. It affects some people more than others but is always noticeable. In one case we created a bit of "safe niche" for the team member until he was through the ordeal. Manager can also help with discreetly letting the rest of the team know. They'll find out anyway, and if you are not comfortable sharing the news, the manager can do it.