Re point number 1 - The Kirk Session Records may have further details if the parties involved were called before them. However these records are not available online but are at the National Records in Edinburgh and at certain local Record centres
Aberdeen City and Aberdeenshire Archives: Aberdeen
Ayrshire Archives: Auchincruive, Ayr
Ayrshire Archives: Burns Monument Centre, Kilmarnock,
Highland Council Archives: Archive and Registration Service, Inverness
Highland Council Archives: Lochaber Archive Centre, Fort William
Highland Council Archives: Caithness Archive Centre, Wick
Orkney Library and Archive: Kirkwall
Scottish Borders Archive and Local History Centre: Hawick
Shetland Islands Council Archives: Lerwick
Stirling Council Archives: Stirling
(from
http://www.nas.gov.uk/about/101101.asp)
(there have been plans from Scotlands People to make the digitised Kirk session Records available through their site but this has been promised for some time now and doesn't look as if it is going to happen within the foreseeable future)
Another option to look for is to check if the childs mother tried to make any court claim against the father - unfortunately again the Court records are not available online.