That's correct. No name is not a blank name, it is no name. Objects that have no name can't "have the same name" as anything.

It will be legendary.

With the original wording of Sakashima the Impostor, there would be a decent argument that it's nonlegendary, as the 'still' in the text would mean the copy effect isn't changing its legendary status.

Unfortunately for you but fortunately for everyone's sanity, the card has gotten an Oracle text revision: You may have Sakashima the Impostor enter as a copy of any creature on the battlefield, except its name is Sakashima the Impostor, it’s legendary in addition to its other types, and it has “: Return this creature to its owner’s hand at the beginning of the next end step.”

When you cast Sakashima, The Sixth Doctor will trigger and copy it. The copy of the spell will be nonlegendary, but when Sakashima the copy enters the battlefield, its copy ability will make it legendary again.

You can get around this to a degree, however, because like most clones, Sakashima's ability is a may. You may choose not to have the copy enter as a copy of anything, in which case it will be a 3/1 Creature - Human Rogue named Sakashima the Impostor. It won't have any abilities, but it also won't be legendary, so it's better than nothing.

Yes.

When you pay 1 life and exile a blue card to Force of Will, you aren't bypassing the cost, you're paying an alternative cost. The new cost to cast the spell is "pay 1 life and exile a blue card from your hand." Tax Collector's effect will add a mana to that total cost you pay.

The difference between this effect and Trinisphere is that Trinisphere will add whatever amount of mana increase that brings the cost up to 3. It will increase the cost to cast Force of Will to 1 life, exiling a blue card, and paying because the casting cost being paid didn't have any mana. On the other hand, if Force of Will were cast for its mana cost, Trinisphere would do nothing since it already costs at least , but Tax Collector would still increase the cost from to .

Tax Collector also isn't symmetrical, while Trinisphere is, so in the second case you'd have to pay to cast that Opt.

Yes. This also works with Fist of Suns. When you go to cast the spell, you choose to pay the alternate cost of . Then near the end of casting you apply cost increases, then cost reductions, where Morophon, the Boundless reduces the cost by . Most of the time this will leave you with a cost of .

And to fill in a bit I missed here, yes, any time it is an artifact again, the Vehicle subtype returns.

This is a weird one, but here you go. TL;DR: No. It'll crew fine. Why this is gets strange.

When Mindlink Mech returns, it'll be an Enchantment. It loses the Vehicle subtype since it's not an artifact, per rule 205.1a: If an object's card type is removed, the subtypes correlated with that card type will remain if they are also the subtypes of a card type the object currently has; otherwise, they are also removed for the entire time the object's card type is removed. It will retain the 4/3 printed in its text box, though this won't do anything while it's not a creature.

Where this gets weird is if you crew it. The first wrinkle is that crew abilities read "this Vehicle becomes an artifact creature until end of turn," which could be problematic since it's not a Vehicle. However, the version of the ability in the comprehensive rules reads thus: "Crew N" means "Tap any number of other untapped creatures you control with total power N or greater: This permanent becomes an artifact creature until end of turn." (rule 702.122a) So the crew ability still works.

When you do, the crew ability's type-changing effect takes precedence over Enduring Curiosity's type-changing effect due to the newer timestamp. But it gets weirder. Becoming an artifact creature is a specific exception to types being overwritten, per rule 205.1b: Some effects state that an object become an "artifact creature"; these effects also allow the object to retain all of its prior card types and subtypes.

So: when the crew ability resolves, Mindlink Mech will go from being an Enchantment with no power and toughness to an Artifact Enchantment Creature - Vehicle that's a 4/3. Then its "first becomes crewed" triggered ability will resolve and it'll become a copy of target creature and its type line will change to that creature's type line plus Artifact and Vehicle (and it'll remain 4/3 with flying). It will most likely lose the Enchantment type. Then at the end of the turn, both the crew and copy effects will expire, and it will return to being an Enchantment with no subtypes.

Yes. Each trigger has an independent chance to copy or return the spell. Copying it with the first resolution and returning it with the second is entirely possible, as is returning it and then attempting to return it again, copying it twice, or returning it and then copying it (which will work, as it will use the spell's last known information).

Copying it and returning to hand (in either order) will result in a single copy of the spell resolving. Ideally you'll hit a return the first time and a copy the second for this scenario, because then you'll get to target the copy already knowing the original won't resolve.

Your high point here is obviously hitting copy twice and resolving three copies, but a copy and return isn't bad, since it means you get the spell off and have it available to cast again.

Yes. Desert Warfare returns 'that card,' and the land is a Desert when it goes to the graveyard, so Desert Warfare will trigger when it goes to the graveyard and will find the card in your graveyard at the end step.