1.) use arrive instead of arrives "When I arrive home, I will call my mom"
2.) arrived is past tense so it should be "When I arrive home, I will call my mom" or "When I arrived home, I called my mom"
3.) I think it's fine grammatically, but the phrase "on the heals of" seems awkward in this sentence
4.) This one is fine
I think #1 (corrected to "When I arrive home, I will call my mom") is the best, but #4 is good too.
There's a useful summary on Grammar Girl, which points out that...
The British traditionally use shall to express determination or intention on the part of the speaker or someone other than the subject of the verb.
But in practice it's a declining usage everywhere - particularly in recent decades, native Anglophones are much more likely to just place heavy stress on will to emphasise "resolute/defiant intention" over and above the standard "future tense" sense of the auxiliary verb.
When Practical English Usage says "shall is not used for decisions", they're talking about the kind of "voluntary, unforced" decision you might make in a restaurant...
1: "I will have the fillet steak" (usually contracted to "I'll have...")
2: "I shall have the fillet steak"
...where the first version is standard, and carries no implications beyond the fact that you've made a choice. The second version (which would be somewhat "unusual", particularly for AmE and younger speakers in general) is only likely to occur in contexts where the speaker for some reason wants to inject a note of defiance/determination (perhaps because whoever's paying the bill has asked his guests not to choose the most expensive items on the menu).
EDIT: There's also this rather quirky BrE "rule": use shall with I/we, and will with you/he/she/it/they (except when expressing determination, in which case the usages are reversed). But as that Oxford Dictionaries link implies, in practice most people don't even know the "rule" (and most of those that do routinely ignore it anyway).
TL;DR: If your coursework specifically addresses "correct" use of shall, and you have an exam to pass, remember and repeat whatever you were taught. If the distinction wasn't covered, or you're not taking an exam anyway, just use will in all cases.
Best Answer
Question 1
Both statements are correct, though they have different emphasis. When you say:
You're simply asking someone to call you. If you add "do", making it:
Then, there's an implication that the other person may not call -- they may be forgetful or unwilling -- and you're emphasizing that you want them to make sure they call you.
Question 2
I speak American English, so I'm going to substitute "will" for "shall", since it's more common in American English. All the following statements are correct and have essentially the same meaning:
The three answers may vary slightly in terms of formality, but it's not a significant difference (shorter answers and answers with contractions are ever so slightly less formal).
Question 3
Only the first sentence is correct:
When constructing a sentence starting with "Does he/she...", you use the base form of the infinitive (e.g. "to do" -> "do").