yes you can!

the way casting spells works is we determine the costs of the spell, and this cost becomes locked in. then we can activate mana abilities to pay these costs. (this locking in is cr 601.2f)

so we put shadow of mortality onto the stack, cost reduce it to 3bb, then pay this cost with our mana sources without our life gain changing the cost.

Even if Semblance Anvil has two cards imprinted onto it, it still only has one cost-reduction ability and so it can only ever reduce a spell's cost by a maximum of 2.

When you start casting a spell, you look at all the card types of cards you have imprinted on Semblance Anvil, and if any of those types match a type of the spell you're casting, then its cost will be reduced by 2.

It works very well, actually!

The cost reduction from Bedlam Reveler’s ability is determined before any costs to cast it are paid, so exiling instants or sorceries to pay for its escape cost won’t make it cost more mana.

So this is the question of the day with about a dozen variations of it hitting my inbox.

When you cast a spell of the chosen creature type, you’ll get Morophon’s discount, but before that happens you’ll have to choose if you’re going to pay any alternate (and/or additional) costs for it. If you choose to pay the alternate cost from Jodah or Fist of Suns, then that sets the spell’s cost at WUBRG. We can then apply Morophon’s cost reduction effect, which will reduce the cost by WUBRG, leaving a nice clean cost of 0 left to pay.

When you cast a spell with X in its cost, you declare the value of X before you apply any cost reduction effects, such as Gargos's ability. What this boils down to is that you get whatever the cost reduction is as a bonus. 

In the case you describe, let's say you have seven mana available to cast Apocalypse Hydra. You could normally cast Apocalypse Hydra with X=5, but once you apply Gargos's ability that would reduce the cost to 1RG, which leaves you with four mana left over. This means that you can cast your hydra with X=9, and Gargos will reduce the cost to the 5RG you can afford to pay.

Yes, this works.

Whenever you cast a creature spell this way, it will be a Hydra spell and so Gargos will reduce its cost by 4. Sounds like a good deal since you’re already up mana after the first spell.

When you cast Everflowing Chalice, you declare how many times you’re going to kick it, then 2 is added to the mana cost for each time you’re kicking it. After that, you’ll apply cost-increasing and cost-decreasing effects to determine the total cost. If you have something like Semblance Anvil that’s making your Chalice cost 2 less, then you can basically kick it once for free. Note that any cost reduction applies to the total cost and not to each kicker cost.