Tout doucement

[formal tulip]

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[smoky narwhal]

« À » is a gendered form. There is
À = feminine
Au/à l’ = masculine
Because the noun here, terre, is feminine, it uses à terre.
The other noun, sol, is masculine, so it uses au.

[quartz sorrel]

hmmm no

à is just the preposition

la is the feminine definite article

le is the masculine definite article

and l' can be either

in the case where à and the definite article "le" are found next to each other, there is an obligatory contraction to "au"

so you have

• à
• au
• à la
• à l'

as for "à terre" vs. "au sol", I'd chalk it up to "à terre" being more of a fixed expression meaning "on the ground"
"au sol" isn't an expression it just literally means "on the ground"

[smoky narwhal]

so it would have nothing to do with terre being a feminine noun?

[opaque rampart]

Nothing at all to do with gender and number.