Is the Russian different than if I wanted to say "I am [currently] going home with Mom?"
No, it isn't. In fact your translation is better than the one offered: e.g. you would not use a uni-directional verb "иду" to describe a repeated or habitual action (you would use "хожу" instead).
P.S. Just like in English, in Russian "Я иду домой с мамой" can describe both an actual unfolding event "I am [currently] going home with Mom" or an intended action, "I am going home with Mom [this is my plan]".
In American English, "I am going home with Mom" can only mean "I am [currently] going home with Mom". If you want to express an intent to go - a plan, or an expression of a future planned course of action - then you'd have to use a future tense, such as "I am going to go home with Mom" or "I will be going home with Mom".
There isn't one. Nor is there one in the English translation I see: " I go home with mom".
This is ridiculous. You can't demand the translation "walk" for some sentences and "go" for others when the verb in those contexts can clearly mean both.
What's the rule about c vs. co? This sentence contains с мамой while a previous exercise contained со мной. I thought adding "o" to prepositions before words with hard consonant sounds.
why is S " with " when a few lessons before S meant " from " ? isn't with " SO " ? if not, why ?
Meaning-wise -- none. "Cо" is the form of "c" that is used in front of some clusters of consonants in order to make it easier on one's tongue. The same goes for к/ко & в/во.