Can I cast a spell after a certain part of the stack is resolved in MTG?
Initially I forgot that Snapcaster Mage has Flash. Oops.

Yes you can do this. The resolution goes like this.


Fireball is cast, put on the stack.
In response, you put Snapcaster Mage on the stack. (Which you can only do, of course, because it has Flash and can be cast as an instant.)
You put nothing more on the stack, your opponent declines to put anything else on the stack.
Because both players pass priority, stack starts resolving in Last In First Out order. Snapcaster Mage resolves.
Triggered ability of Snapcaster Mage is put on the stack. You choose Mana Leak as the target.
Both players pass priority. Last effect on stack resolves, Mana Leak gains flashback.
As Mana Leak is an instant (IMPORTANT) you can now pay the flashback cost (in this case same as the casting cost) to put Mana Leak on the stack.
Both players pass priority. Mana Leak resolves.
Your opponent must now pay {3} or Fireball is countered.


The important bit to remember is that yes, as usual, you can put spells/effects on the stack at any time even after spells have started to be resolved, as long as said spells/effects can be cast at "instant speed", not "sorcery speed".

The comprehensive rule which answers your question "Can I really do this after the resolving of spells on the stack has begun?" is 116.3. which determines when players have priority and I will paraphrase:


Active player gets priority at the beginning of steps and phases (after "beginning of" triggered abilities are put on the stack.)
Active player gets priority after a spell/ability on the stack resolves.
A player gets priority after they put a spell/ability on the stack.
If a player passes priority, the next player in turn order receives priority.


So the priority rules make it clear that there is always a chance to respond even after spells on the stack start resolving.