Raising the pH helps with the detergent action, also the washing soda "softens" the water - calcium and magnesium in the water form carbonates, replacing the sodium in the washing soda.

You probably could, but IMO that's an excessive amount of borax in the recipe. Most washing mixes only use a tiny amount of borax, it's mostly soap/detergent and sodium carbonate (washing soda)

Also the "extra" carbonate group is dislodged as carbon dioxide:

2 NaHCO~3~ (baking soda) → Na~2~CO~3~ (washing soda) + H~2~O + CO~2~

Baking soda is NaHCO~3~ , Washing soda is Na~2~CO~3~

Baking soda is (incorrectly) called "Bicarb" because there is twice as much Carbonate (CO~3~) per Sodium (Na). Heating it causes some to be released as water and carbon dioxide:

2 NaHCO~3~ → Na~2~CO~3~ + H~2~O + CO~2~