No.1 is, in my opinion very badly written...
Probably we shall be astonished at the number of propositions met with in everyday life which we shall find it necessary to class as groundless.
We will probably be amazed at the number of propositions we meet with in everyday life where we will find it necessary to class them as groundless.
Better, but still a bit of a mouthful.
Many of our beliefs are held simply as a result of the fact that we happen to have been "brought up" with them.
Brought up means raised as a child. You bring up children, but children are brought up.
You could be brought up to believe in certain things
To bring up to somebody, means to raise an issue with them, such as 'I will bring that up with him when I see him'
You interpretation is correct. Just because something has been around and accepted for a long time doesn't mean it is correct.