A flag-packing problem

^{An entry in Fortnightly Topic Challenge #45: Flags}

You are provided with a 9x9 grid of squares and 21 minimalistic flags (pictured below, all shown to scale).

TASK: Assign a colour (Black, Blue, Green, Orange, Red, White, or Yellow) to each of the 81 numbered cells so that the grid can be exactly partitioned into 21 shapes equivalent to the pictured flags (without rotation or reflection). Each cell must be covered by part of exactly one flag, every pictured flag must appear in its own right, and the following rules must be satisfied:

1. Horizontally or vertically adjacent cells belonging to different flags must be different colours.

2. All cells containing a square number are the same colour.

3. Only 1 cell in the bottom row of the grid is occupied by a European flag (i.e. one of the flags in the top row of the image, from Denmark to Russia).

4. The flags of Austria and Russia are adjacent (i.e. at least one cell in the Austria flag borders a cell in the Russia flag, horizontally or vertically).

This puzzle can be solved purely by logical deduction - no guesswork is required. Please explain the logical steps leading to your answer.

NB The symbols on the flags of Cameroon and Guatemala can be ignored. Flags are all 3x3, 3x1 or 1x3 grid cells in size.

_{Colour guide available here. Bk=Black, B=Blue, G=Green, O=Orange, R=Red, W=White, Y=Yellow.}