Is there any software for Windows that can monitor the activity of my children - what pages they are visiting, what they type, what passwords they use etc?
I'm looking for something that's hard to detect and delete/disable, and can be permitted by antivirus software (or simply not detected by it).
I don't want to restrict the pages in advance as I don't want them to know I'm watching them.
If they have admin privileges, then there is nothing stopping them from killing the keylogger.
Plus getting a keylogger and parsing through the files everyday takes time.
This is time better off spend talking to them and gaining your trust.
If you think their internet surfing habits are a nightmare, then wait until they get out of the house and actually act on those wishes.
You do want to restrict the pages in advance, to protect them from content they should not get to see.
Surveillance / keyloggers of which they don't know is a breach in their privacy, even though they won't know it can end up painful for them the moment they found out that you watched them in private. When they do something wrong then you didn't protect them of visitting the site (now they are shocked), only to tell them that you watched them afterwards (now they are shocked again).
Don't just let things happen, take preventive measurements. It's like watching your child fall of a ledge because you didn't place a fence, if you place the fence it didn't happen in the first place...
If you want to know what they are doing, spent time with them or visit them occasionally...
There are always their logs and history if you do want to check on them, but don't confront them directly.
If this is a desktop computer that will not be leaving the house, will always connect to the home internet connection. Set your router's DNS to the OpenDNS service and set up an account. You can then get monitoring reports of each and every website they've been to and select levels of filtering if you want.
Every computer on your network will use the DNS addresses specified by the router, and the filtering will be completely transparent to the kids.
If you want to see what they're typing on those sites, you'd need a keylogger or parental control software of some sort. The most important thing there is that your kids only be able to use limited user accounts and that you don't ever give them the password to the admin account with which you'd control the computer.
Regarding the monitoring software coexisting with your virus software: You shouldn't try to find a tool that can hide from the AV, you should find a tool that's legitimate and thus can coexist with the AV software without raising alarms. Most AV software allow you to choose applications to "whitelist" so that even if the software behaves in a way the AV software thinks is suspicious, it will leave it alone.
UPDATE:
Making a house rule that all computers are always kept in the living room or dining room or den or other open and public room that everybody has access to all the time can be as good a filter and monitor as anything else.
