From the beginning of the battle phase to the end, when can each player cast spells or quick-spells, and activate traps?
Short Answer: If the card is Speed 2 or higher basically anytime, unless otherwise restricted by specific conditions or requirements.
Longer Answer: Basically anytime, although this also depends on the specific restrictions of each card, and the conditions that have to be met in order to be able to activate them.
As explained in previous questions, only Spell Speed 2+ cards can be activated during the Battle Phase(that is, normal trap, counter trap, continuous trap, quick-play spell, or any Speed 2 monster eff). Also remember that if you set one of these cards you can only activate it on the next turn, so be careful when doing so.
If your card is one of these you can activate it, unless otherwise restricted. For example, a card like Mirror Force can only be activated "when an opponent's monster declares an attack", and not in other situations:
When an opponent''s monster attacks, negate the attack and destroy all opponent''s monsters in Attack position.
In this case only when an attack is declared (during attack declaration) is that you can activate this card. If Attack Declaration passes, and then you enter the Damage Step, you cannot activate it anymore, as you would have lost timing.
Let's analyze another case, Rush Recklessly:
Target one face-up monster on the field; it gains 700 atk until the end of the turn.
This one has more freedom of when to activate. This card you can activate anytime during the Battle phase, even during Damage Step (as any card that modifies Atk or Def stats can be activated in that step, as indicated in some rulebooks).
Be aware that this is not the same as Damage Calculation, where only some cards can be activated (like Blackwing - Kalut, the Moon Shadow) You could even activated it at the start of the BP... but you would achieve greater tactical advantage if done on Damage Step (as your opponent will have attacked already thinking his Atk was higher).