what are the limits of c++ progamming for the uno & gsm

This is a purely hypothetical/theoretical limitation question.

I'm trying to understand what I can, and can't do with the code and the gsm module + uno.

I'd like to do a few things:

  1. if I text the GSM module, have my code store the new phone number(s). Instead of hard coding them and uploading it to the board, I'd like to text the arduino to add additional numbers and have it save.

I think it's possible if I have some function that looks for a keyword in a message receipt, like if sms_recieve("AddNumber",+112345678)
Update_Numbers_list; ? Something like that? Just want to know if it's possible.

  1. logging. Is it possible to have debug logs stored in anyway?

Basically, is it possible to modify/add the existing code on an Arduino purely through texts if I program it in a way that looks for these keywords and strings and then executes the code to update itself if it receives these messages?

I'm not sure this is what you're looking for, but...

You can buy SD shields that plug into the Uno and can be used to store data. You can then retrieve that data and display it on an Arduino display device. For example, some TFT displays have SD card slots build into them. It would not be hard to receive data from an external device, store it on an SD card, and then write a UI do display it as needed.

Logs usually go on an SD card, so if you go that way you will want an SD card shield.

Yes, of course the arduino can accept incoming texts and parse them. Most arduinos also have a small amount of static EEPROM memory, which is idal for small amounts of data like a list of phone numbers.

As to how to go about doing it:
a) work through the examples, learn how to program; or
b) offer an inducement of some kind on the "Gigs and Collaborations" subforum.