Putting it all together
Tumbler41's answer figured out the majority of this (make sure to upvote that one). Comments from kayzeroshort and M Oehm filled in some more.
Each image corresponds to a series of letters, or mnemonic, and the position of the image in the password indicates which of those letters to use.
Here are the mnemonics for each symbol:
Stands for: Huron, Ontario, Michigan, Erie, Superior
Meaning: The five Great Lakes on the Canada/U.S. border.
Clue: "I should get a boat" hints that this is referring to lakes.
Stands for: Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune
Meaning: The order of the planets in our solar system, starting from the sun.
Clue: "Alas, no pizza" refers to an earlier version of this mnemonic, which used "My very educated mother just served us nine pizzas" to include Pluto as well. Since Pluto was demoted to planetoid, the "pizza" is no longer there.
Stands for: Parentheses, Exponents, Multiplication, Division, Addition, Subtraction
Meaning: This is the order of operations for mathematics. It tells you what part of an equation to evaluate first (from left to right).
Clue: "Order matters here" refers to the order of operations.
Stands for: Red, Orange, Yellow, Green, Blue, Indigo, Violet
Meaning: These are the colours of the visible light spectrum, from longest to shortest wavelength.
Clue: "Frost me once..." -- The response would be "Frost me twice", which is an oblique clue to Bifröst, a rainbow bridge in Norse mythology.
Stands for: numbers/symbols, because it is itself a symbol
Meaning: This is the symbol that was used by the artist Prince to refer to himself for a while, instead of using a name.
Clue: "A true artist" points people to the fact that this is a symbol for an artist, in case they had not made that connection.
Stands for: Good Afternoon/Morning/Evening/Night
Meaning: This is commonly used in The Sphinx's Lair as a greeting, because there are people there from so many different time zones.
Clue: "Good Morning?" gives the clue that this may be referring to different times of day.
Stands for: Kingdom, Phylum, Class, Order, Family, Genus, Species
Meaning: This is the taxonomic order, in which all living organisms are classified, from the most general (Kingdom) to the most specific (Species).
Clue: "Family friend" is a clue to the different groupings, which include Family. "Order matters" indicates that this is the taxonomic order.
Stands for: North, East, South, West
Meaning: These are the directions on a compass rose,
indicating the four cardinal directions.
Clue: "Don't look down" indicates that you should be looking in other directions.
Stands for: B, E, A, D, G, C, F
Meaning: This is the order in which flats appear in music. A piece with a single flat note will have B$\flat$, while a piece with two flats will have B$\flat$ and E$\flat$, etc.
Clue: "Greatest common factor" is often abbreviated GCF, which are the last three notes in the above mnemonic.
These are different from the others. As the clue indicates, moving them around doesn't matter. They are simply modifiers for the other components of the password. A digit indicates that the character in that position of the password should be uppercase.
So a password containing the digits $3$ and $5$ would have the 3rd and 5th characters in uppercase. (An "uppercase" digit is assumed to be the associated symbol on the keyboard.)
The final answer
Using this information, we can easily solve the final password:
It will be:
- h (the first characters of HOMES)
- e (the second character of PEMDAS)
- y (the third character of ROYGBIV)
- 4 (because the PRINCE symbol is in the 4th position)
- n (the fifth character of GAMEN
- g (the sixth character of KPCOFGS)
- 7 (because the PRINCE symbol is in the 7th position)
- 8 (because the PRINCE symbol is in the 8th position)
- 9 (because the PRINCE symbol is in the 9th position)
But because we have the digits $148$ included as well, we "uppercase" the 1st, 4th, and 8th characters, to give us the final solution of: