I think I finally have this, and it's taken me an embarrassingly long time to realise what's going on here (given how I spent most of my childhood)... but I believe the grid above should be completed a little something like this:
Because these 'crossword clues' all actually point towards...
...the official names given to particular types of Lego brick!
1A Catan resource = BRICK
4A Cables used to restart car batteries = JUMPERS
5A A piece of Earth's lithosphere = PLATE
1D To apply backsplash, for example = TILE
2D Grates on the front of automobiles = GRILLES
3D Technique, or a technicality = TECHNIC
'Bricks' are the basic building blocks of Lego, which come in a number of sizes. 'Plates' are flatter pieces, one third of the height of a brick, while 'tiles' are like plates but without any studs on top, and 'jumpers' are like plates with the studs offset by half (so-called because they allow you to 'jump' half a unit). 'Grilles' are tile-like pieces with slots cut out along their length, and 'Technic' bricks have holes passing through them.
The full assembly of the 'grid' can probably best be seen in an angled shot like this:
Note that my 'brick', 'plate', 'tile' and 'technic' lines have had to be made from multiple pieces of those types, as it is not common to find 1x5 dimension pieces in these types. The 'grilles' and 'jumpers' are pluralised in the crossword clues because the most common standard size for these tiles is 1x2, hence multiple would be needed to complete their lines in the diagram.
Note that this entirely explains why the row for 4A is offset strangely. Also, the '2x2' in the title probably reflects...
...the dimensions of the stereotypical generic Lego brick:
The black spaces of the grid even resemble this somewhat! Altogether, this theme makes perfect sense of Hint 3: "Piece by piece, someone surely has to be putting this together..."
