1. If I name the wrong company name (because I can't find it), can I sue later if I find out the real name in court?
2. Also, I think I will be naming the individuals as well, in case it is not Incorporated.
Will this be OK?
3. Is there any way to find out a company name other than the Ministry of Finance, they searched every name I had (3) and said it was not registered. Owner tells me it is, and letterhead say INC.
4. I have a document from the owner that went to court 2 months ago and in that document she says her company name, as I know it, and she says it is registered with min of finance (all in this court document) Is it an offense to lie if you are not INC?

Every State/Province is different and I can only address what I know BUT:

Suing wrong name - named parties would appear; if they are incorrect Judge would dismiss against them and in every case I've ever seen obtain the correct name/names (as part of questioning concerning why they are the wrong people) and you would be provided with that information. You would then refile against the correct names.

Suing wrong individuals - as long as you are not naming people to harass them; as long as you think you have the correct parties you are OK.

Ministry of Finance - you are in Canada, I am in US so I do not know.

Company name - yes, in the US it is an offense to say you are incorporated if you are not.

Hope this helps a little -