** SPOILERS **
This post will reveal the entire worked solution to this riddle.  If you're still working on it, stop reading here!

The Riddler is pretty explicit about how to solve this one: “a baffling set of thieves' tools, which she always uses once each.”  The instructions are labeled “tools,” so we have to use them once each to find the answer.

Finding the starting point isn't hard, since one of the tools tells us how to find it explicitly: “The safe is named after a Russian fairy tale about a girl named Marusia.  The starting code is the full English title of that fairy tale, ignoring spacing.”  Google or Wikipedia will quickly identify the fairy tale: it is called The Fiend.  So the starting code is THEFIEND.

From here, it's a matter of applying the tools in an order that fits the code at hand as we gradually transform it.  If we properly understand our tools, we'll generally find that at most two of them apply to the code, and making an incorrect choice leads to a dead end quickly.  For example, when it comes to the starting code THEFIEND, we find that the first tool (“Anagram both codes . . .”) doesn't apply, because we only have one code.  The second tool (“Caesar shift the resulting code . . .”) also doesn't apply, because this is our starting code and doesn't result from anything.  If we work down the list, we find that the only tool that applies is the one that begins “Split the code.”  Continuing in this vein, with some trial and error, we can work out the puzzle like so:

The safe is named after a Russian fairy tale about a girl named Marusia.  The full English title of that fairy tale is the starting code.
THEFIEND
       
Split the code into two codes: one made up of the consonants in order, the other made up of the vowels in order.
THFND | EIE   

Move the fourth and first letters of one code to the first and third positions of the other, respectively; you will see a mesh.
HFD | NETIE  [A net is a mesh.]

Insert GO and AT to form a word.
HFD | NEGOTIATE

In one code, spell out the first letter as its Greek lookalike; in the other code, rewrite the last three letters as a soundalike numeral.
ETAFD | NEGOTI8  [H is the Greek letter eta; ATE sounds like 8.]

Anagram both codes into words; leave the numeral where it is, but halve it.
FATED | TOEING4

Change out Fresno-Yosemite International for Leeds-Bradford International, and Tozeur for Oulu.
LBAED | OULING4  [These are all airport names.  Their three-letter IATA codes are respectively FAT, LBA, TOE, and OUL.]

Switch the verb suffixes between the codes.
LBAING | OULED4

Combine the codes into one by using the letters of the first in the odd positions, in order, and the letters of the second in the even positions, in order.
LOBUALIENDG4

Delete the longest word, and insert the title of the Katy Perry song it reminds you of before the numeral.
LOBUDGET4      [The longest word was ALIEN.  ET is a Katy Perry song about sexing up an alien.]

Switch out the longest word for the government agency responsible for preparing it, briefly.
LOOMB4  [The longest word was BUDGET, the responsibility of the Office of Management and Budget, or, “briefly,” OMB.]


Replace the initial four-letter word with a word that can go before it.
HEIRB4  [The initial word was LOOM.  HEIR is a word (in fact, the only common word) that can go before LOOM.]

Increase the atomic number of the noble gas by 4, and of the metalloid by 24.
CIRCU4  [The noble gas and metalloid are the chemical elements helium and boron: HE and B from the previous code.   The atomic number of helium is 2; if you increase it by 4 you get 6, the atomic number of carbon, abbreviated C.  The atomic number of boron is 5; if you increase it by 24 you get 29, the atomic number of copper, abbreviated Cu.  So we switch HE for C, and B for CU.]

Replace the name of the doctor with its mana cost.       
2UB4  [The name of the doctor is CIRCU.  Circu, Dimir Lobotomist is a Magic: the Gathering card, and its mana cost is 2UB.]

Reverse the order of the four characters.
4BU2

Increase the numeral of the rock band by two.
4BU4  [U2 was the rock band.]

Change both Arabic numerals to Roman numerals.
IVBUIV

Caesar shift the resulting code eight backward.
ANTMAN

And the result, ANTMAN, is the riddle solution.