As you guess, the use of the definite article with nouns that carry some kind of emotional attachment (like child or mother) sounds a little impersonal. Actually this is the intended nuance. The definite article removes the emotional attachment, allowing the speaker to talk as an unaffected observer:
During the brief war, the father of the family left to fight with the partisans. They never saw him again.
Of course, it's not always this dramatic, and as the reader you may still feel some kind of emotional response, in the appropriate context.
This kind of thing is common in scientific writing where "the subjects" of the study are meant to be observed and measured impartially and objectively. In this context there is no negative connotation of "the mother" or "the child". It's merely a statement that expresses certain facts.
A native speaker could use either of those sentences. "Is there another day when you can come in?" has better grammar. In fact, the other one is bad grammar, but it's commonly used by native speakers.
Sentence 1: A relative clause with "when" describes a noun by using that noun in an adverbial of time. In this case, the deeper structure of the sentence is:
"Is there another day?" + "You can come in on that day."
The pronoun "when" replaces the adverbial "on that day". "Day" refers to the same "day" as earlier in the sentence.
Sentence 2: A relative clause with "that" describes a noun by using it as a subject or an object in another sentence. The grammar doesn't work here, because "You can come in" already has a subject, "you", and there's no place for an object. The word "in" is part of the phrasal verb "come in", not a preposition, so it cannot take an object. "...that you can come in" suggests a deeper structure like:
*"Is there another day?" + "You can come in day."
which is nonsense. One way to fix this is to add the preposition "on" to the end:
"Is there another day that you can come in on?"
You can come in on a day, so this is good grammar.
Best Answer
surface is the least appropriate term to use: it really refers to the upper or outside layer of something. You could talk about the entire surface of the cuboid (all six faces) as one surface.
For side, even the major dictionaries disagree about what it means:
So, a cuboid could have either two sides (left and right) or four sides (left, right, front and back).
In everyday speech, face is used to describe the front of an object, however in geometry it is used in the way that you require- for example:
All of these terms are therefore prone to some confusion, but if you wish to describe each of the six faces using the same term, face is probably the best term to use.