Problems restoring from a time machine backup - can I replace apps manually?

You can drag and drop most apps across. Some apps when installed placed some files in other areas to work correctly and will need to be reinstalled. I wouldn't copy of the entire Library folder over as it could likely cause some problems. You can pull out some specific folders within the Library folder though. For instance, app preferences are stored in ~/Library/Preferences/ and you may have some important things in ~/Library/Application Support/. You likely won't be able to tell which apps need reinstalling until after you try opening and they fail to open or give errors.