Can I tap a land to add mana to my mana pool, use Vizier of Tumbling Sands' ability to untap that land, then tap it to add another mana to my mana pool and finally cast a spell?
Following on from that, a fairly simply question: when is one actually allowed to tap a land for mana? Is it only when one has cast a spell and then has to pay for it?
I'm trying to find the answers for these in the rules but am battling!
Can I tap a land to add mana to my mana pool, use Vizier of Tumbling Sands' ability to untap that land, then tap it to add another mana to my mana pool and finally cast a spell?
Yes, perfectly fine.
Following on from that, a fairly simply question: when is one actually allowed to tap a land for mana? Is it only when one has cast a spell and then has to pay for it?
Most mana abilities are activated abilities without further restrictions, including those of lands, so you can use them whenever you have priority, essentially whenever you can cast an instant. In addition, mana abilities (without timing restriction forbidding this, like Lion's Eye Diamond or Grinning Ignus) can be activated whenever you are asked for a mana payment. Be that during the casting of a spell, activation of an ability, during attacker declaration due to a Ghostly Prison, or during the resolution of a spell/ability (eg. Mana Leak).
I'm trying to find the answers for these in the rules but am battling!
You may want to look at rule 106. Mana and 605. Mana Abilities.
In general, you can activate an activated mana ability (which includes tapping a land for mana) "whenever [you] have priority, whenever [you] are casting a spell or activating an ability that requires a mana payment, or whenever a rule or effect asks for a mana payment" (C.R. 605.3a, 106.12). Thus, for example, if you have priority, you can tap a land for mana, then cast such a spell or activate such an ability (such as the first ability of Vizier of Tumbling Sands) without necessarily spending that particular mana to pay for that spell or ability.
You can tap a land for mana anytime you have priority. When you do this without directly casting a spell it's called floating a mana. It's often used the way you described for using Vizier of Tumbling sand to add mana or in the current standard format Wilderness Reclamation untaps all lands at the end step so people will go to the end step and let it trigger then tap all their lands for mana, let them untap and then tap them again to effectively double their mana and cast large X spells like Expansion//Explosion
Following on from that, a fairly simply question: when is one actually allowed to tap a land for mana? Is it only when one has cast a spell and then has to pay for it?
I'm trying to find the answers for these in the rules but am battling!
Yes, perfectly fine.
Most mana abilities are activated abilities without further restrictions, including those of lands, so you can use them whenever you have priority, essentially whenever you can cast an instant. In addition, mana abilities (without timing restriction forbidding this, like Lion's Eye Diamond or Grinning Ignus) can be activated whenever you are asked for a mana payment. Be that during the casting of a spell, activation of an ability, during attacker declaration due to a Ghostly Prison, or during the resolution of a spell/ability (eg. Mana Leak).
You may want to look at rule 106. Mana and 605. Mana Abilities.
Former Rules Advisor
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EDIT (Oct. 31): Clarification.
EDIT (Jun. 6, 2021): Correctness edit.