The text is clear: the spellcaster must a) be able to cast at least one of the spells in the staff and b) possess a slot equal to or greater than the highest-level slot. Most of the time, this means they must be able to cast any of the (one or more) highest-level spells in the staff.
Occasionally, a multiclass caster might be able to satisfy both conditions with casting from different classes. A Wizard 1/Cleric 13, for example, can of course give up a (Cleric) 7th-level slot for DBF, and can* cast burning hands: boom, mission accomplished, staff can be recharged.
(For clarity, this is most similar to what's written as interpretation 2, except that there is no requirement for the character's highest-level slot, only a slot that matches the staff's highest-level spell. A Wizard 17 with such a staff would not need to sacrifice one of their precious 9ths.)
*With a suitable entry in their spellbook, anyway
The ring of mystic fire only affects the upcoming spell once
The ring of mystic fire (Magic Item Compendium 125) (7,500 gp; 0 lbs.) does, indeed, grant "a bonus to the amount of damage you deal with the next fire spell you cast before the end of your turn," but the ring only grants that bonus once. Each ray of a scorching ray spell isn't the spell; instead, the whole spell creates multiple rays. The ring improves the spell's damage only once by a maximum of +4d6 no matter how many effects result from the ring-modified spell.
(Ruling that each ray of the scorching ray spell benefits from the ring multiplies the power of the ring by the number of rays—with rather extreme consequences at low levels, especially given that the ring increases also the wearer's caster level for fire spells!)
Although the DM has several options available to adjudicate the ring's effect—allowing the player to distribute the total number of extra dice damage the ring grants among the rays before (or even after) the rays're launched, for instance—, this DM would keep things simple and have the player add the extra damage to his PC's first ray in a volley from the spell scorching ray, in much the same way that precision damage applies to the first attack in a similar volley (Rules Compendium 42). (In fact, I have a fire wizard in a current campaign who's eyeing this very item, and that's how I'm ruling it works.)
Hence, for example, under such a ruling, a level 6 caster wearing a ring of mystic fire can take a swift action to spend 3 of the ring's charges. Then the caster can cast scorching ray, creating 2 rays (because of the increase in caster level granted by the ring) and making 2 ranged touch attacks. If the first ray hits, it deals 8d6 points of fire damage. If the second ray hits, it deals 4d6 points of fire damage.
To answer your question: charges on weapons (presumably magical ones based on the question) are regained depending on the item. If you are using any of the items in a Hardcover book such as the Dungeon Master's Guide or Xanathar's Guide to Everything, then it should say how many charges the item has, and when those charges are regained. If the item stats do not mention charges, the item doesn't have charges. If it mentions charges and it does not say when it regains charges, assume the item does not regain charges.
That being said, for the most part, magical items regain a certain number of charges daily at dawn. The DMG does mention, however, that the DM can perhaps arbitrate a different time depending on the item (for example, a sword that is "forged on the moon" might regain its' charges at Midnight, or perhaps may only regain them once a month when the Full Moon is out). I cannot find the page number but I do recall reading this.
Moving along, as I said earlier, most items that have charges do specifically say when the charges are regained.
Now, in D&D 5e you do not take turns or actions to rest. The DMG specifies two general kinds of rest, short and long. Short Rests are 1 hour in the RAW and Long Rests are 8 hours. If the magic item is somehow dependent upon the player's resting schedule for regaining their charges, then this is what you would refer to.
Finally, magic: there is no spell, cantrip, or other magical effect (that I know of) that makes magical items regain their charges more or less quickly, excepting the open-ended version of the Wish spell which could theoretically be used to restore any number of, or all, charges to a magical weapon or other magic item that has a limited number of charges. That being said, however, your DM may choose to add or may allow you to add a spell that has such an effect, but you should discuss this with them as this would fall under house rules.
Apologies for not referencing the staff of fire specifically as I wrote this out before that edit was made.