I'm not sure what the connection is between Lettice Knollys and Cornwall as Lettice was born in Rotherfield Greys in Oxfordshire then lived in Chartley, Staffordshire with Walter Devereux, her first husband, and then London with her second husband, the Earl of Leicester (Dudley). I haven't found any mention of Cornwall at all in connection with Lettice.
The poet Sidney was Sir Philip Sidney from Penshurst in Kent - see http://www.infobritain.co.uk/Sir_Philip_Sidney_Biography_And_Visits.htm for more details on him. His mother was Mary Sidney who, as you say, was Robert Dudley's sister and also sister of Guildford Dudley, Lady Jane Grey's husband. So, no, not Sidney Godolphin.
When I googled Lettice Knollys portrait and St Michael's Mount, I found a page which said that during Elizabeth I's reign the Mount and two manors were sold to Sir Robert Cecil, but I'm not sure how Lettice's portrait got there, perhaps it was just part of the National Trust collection and they decided to hang it there.
Sorry not to be of more help!