Q.I.) Why different authors assign different characters to the same orbital or normal vibrational mode? What is the source of this differences?

If different orientations of the molecule are used, symmetry labels corresponding to the same orbital or mode may be different.

For example for the water molecule which symmetry axis coincide with Z axis, we still have two choices for the molecular plane: XZ and YZ, presented on the picture below.

Two different orientations of water molecule

Following table shows symmetry labels of water normal vibrational modes for these two different orientations.

Mode	Picture	Irrep for XZ	Irrep for YZ
Asymmetric stretch		b1	b2
Symmetric stretch		a1	a1
Bending		a1	a1

The difference in labeling for asymmetric stretch directly follows from the definition of irreps for the C_2v point group given in the table below.

As we can see b₁ and b₂ irreps differ in the way they change sign on reflections in XZ and YZ planes. Thus different selections of molecule plane (XZ or YZ) lead to different labeling of asymmetric stretch (b₂ or b₁).

In order to solve these ambiguity problem the symmetry conventions were developed.

Q.II.) What are the conventions for symmetry notations in computational chemistry and spectroscopy?

Conventions of the symmetry notations are given in the paper:
R. S. Mulliken,. J. Chem. Phys., 23, (1955) 1997

The most important symmetry conventions are summarized below:

• Lower case letters should be used to describe normal vibrational modes and orbitals. E.g. a_g normal mode
  

  Upper case letters should be used to describe vibrational, vibronic and electronic states. E.g. A_g electronic state

• Character symbols and their definitions should be used exactly as given in G. Herzberg, "Infrared and Raman Spectra of Polyatomic Molecules (New York, 1945), besides T and t symbols should be used instead of F and f for triply-degenerate species.

• For the following symmetry point groups molecules should be oriented as describe below:
  1) planar C_2v molecules: X-axis perpendicular to the plane of molecule, Z-axis is axis of symmetry
  2) planar D_2h molecules: X-axis perpendicular to the plane of molecule, Z-axis passes through the greatest number of atoms. If the last conditions is not unambiguous Z-axis should pass through the greatest number of bonds. For other symmetry point groups read the original paper.

• Normal vibrational modes should be labeled following next rules:
  1) Modes should be grouped in blocks with the same character.
  2) Blocks with different characters should be ordered as in Herzberg's book (p. 272)
  3) Modes inside of each block should be ordered from the highest to the lowest

Q.III.) What information should be presented to make symmetry assignment unambiguous?

The only way to make the symmetry assignment unambiguous for the reader is either to follow standard conventions (preferred) or to specify orientation of the molecule in space.