The Advancing Companions section in the Pathfinder System Reference Document seems to indicate that the answer is yes on all counts:
Another issue is who gets to control the companion's advancement. Animal companions, eidolons, and cohorts all advance much like PCs, making choices about feats, skills, special abilities, and (in the case of cohorts) class levels. Whoever controls the companion's actions also makes decisions about its advancement, but there is more of a shared role between you and the GM for some types of companions.
[...]
Cohort: A cohort is generally considered a player-controlled companion, and therefore you get to decide how the cohort advances. The GM might step in if you make choices that are inappropriate for the cohort, use the cohort as a mechanism for pushing the boundaries of the game rules, or treat the cohort unfairly.
(emphasis mine) and from Aspects of Control:
Sentient Companions: a sentient companion (a creature that can understand language and has an Intelligence score of at least 3) is considered your ally and obeys your suggestions and orders to the best of its ability. It won't necessarily blindly follow a suicidal order, but it has your interests at heart and does what it can to keep you alive. Paladin bonded mounts, familiars, and cohorts fall into this category, and are usually player-controlled companions.
And in Monster Advancement.
Adding Racial Hit Dice: Adding racial Hit Dice to a monster is a similar process to building a monster from scratch. As additional Hit Dice are added, other abilities increase in power as well. Additional Hit Dice usually results in better attack bonuses, saves, hit points, and skills, as well as more feats. It can also include additional spellcasting capability and other powers.
All but the last of these passages can be found in Ultimate Campaign. The last passage is part of the Bestiary, on page 295.
Since what spells a faerie dragon knows are a decision it makes as it advances in racial HD, and since you, not the GM, control such advancement, and, in fact, determine all aspects of your familiar with only minor oversight (as shown above), you can certainly swap out a faerie dragon's spells known as you generate it, just like you can its skill points, feats, and all other decision-based aspects of its character.
As for item usage, the dragon casts spells "like a sorcerer", so for items which benefit from or require sorcerer spell ability (like a wand of Fireball or a Staff of Meteor Swarm or, like you mention, Pages of Spell Knowledge) the dragon qualifies as a sorcerer. However, it is not a sorcerer and so items that rely on it being one (class requirement items) or possessing other class powers (Robe of Arcane Heritage) would not function.
You are right that the spellcasting does not increase as you (the master) level up, but if the familiar were a wild faerie dragon it would increase with additional racial Hit Dice, as per the normal rules for advancing monsters with spellcasting ability.
Edit: This answer predates several rulings and clarifications made by WotC and Crawford in particular. I'm leaving it in place for historic purposes, but it's no longer a particularly useful answer.
Strictly speaking, there is no clear interpretation. All three cases are justifiable. Also note that 5e discourages literal "rules as written" meanings. As the designers have repeatedly said: "rulings, not rules." The rules were explicitly not written to be scrutinized as a lawyer scrutinizes the law, so we should not be surprised when the end result of "it's ambiguous" is what we find.
Firstly, "natural" melee weapons are, as far as I'm aware, considered melee weapon in 5e. [ See also.] There is no distinction between a mace and a hoof as far as "counts as a weapon" is concerned in 5e. I don't know if this is explicitly stated anywhere (I thought it was) but Unarmed Strike is explicitly listed as a weapon on the weapons table, and it's strongly implied since all monster stat blocks say things like "Bite Melee weapon attack: [...]". As far as I can tell, if you make an attack with it, it's considered a "weapon" in 5e. Something is a weapon if it's used to make an attack, then, not because it's got a weapon tag on it.
You could argue a Case 1 by saying that find steed only modifies the target of the spell. The spell still refers to "you," so even though it effects your mount, that extension does nothing. In other words, you argue that for Range: Self spells, "you" in the spell description means exactly, "you, the spell caster," and never, "you, the spell's target." This interpretation, however, also modifies spells like divine favor, detect evil and good, crusader's mantle (that one's a bit of a pickle to decode with a mount), and every other Range: Self spell. I'd argue it's all or nothing here. Either they all work on the mount (in some way), or none of them do. It doesn't matter how you rule here, but you should be consistent. Given the number and range of Paladin spells that are Range: Self I question an interpretation this narrow as being the design intent, but it's certainly supportable. About the only thing that reinforces this interpretation is the fact that the Smite class ability does not work on a find steed mount, but that's only because the class ability isn't a spell so it doesn't qualify for find steed's expansion.
The difference between Case 2 and Case 3 is deciding if find steed changes the wording of spells to "The first time both you and your mount hit with a melee weapon attack [...]" or changes to "The first time either you or your mount hit with a melee weapon attack [...]". Honestly, there's not enough information to decide either way. The spells are not written with find steed in mind, and find steed is not worded to make the end result clear.
You can argue Case 2 by saying, "The spell is intended to only affect a single melee attack; if it were intended to affect multiple targets, it would be higher level or otherwise deal less damage."
You can argue Case 3 by saying, "Find steed, like find familiar or hunter's mark, is a class ability masquerading as a 2nd level spell, and that wording was put there to have an intended effect. Furthermore, making a Paladin more deadly while mounted -- a fairly rare situation in most campaigns, IMX, and small Paladins are already less threatening -- is in-line with the desired result of the theme and flavor of the class. Given also the relative scarcity of spell slots, the additional power is probably not significant in most cases." This is not a particularly crunchy argument, but given that 5e does not separate crunch and fluff, it is legitimate.
If I were to rule conservatively, I would probably rule Case 2. If I were in a more liberal frame of mind, Case 3 would be reasonable. As it stands, I don't see any compelling justification for any one interpretation.
Best Answer
No, this will not work
Twinned Spell
(emphasis mine)
So, when you use Twinned Spell you are targeting an additional creature thus conflicting with the one target restriction for that effect in Find Steed:
(emphasis mine)
The same is true in the reverse case as well; using the spell sharing feature of Find Steed also results in multiple targets thus making it invalid for use with Twinned Spell.
Jeremy Crawford seems to support this as well in his description of Twinned Spell:
Official Ruling
Jeremy Crawford made an almost identical ruling for the case of split enchantment (a wizard enchantment ability)
Split enchantment says:
This is an almost identical case to the case of Find Steed and Twinned spell. Thus the ruling absolutely applies here as well.
Both of these Spells change a one target spell into a spell that targets two creatures. No matter how you cut it, after using either of these abilities, you now have a spell that is targeting more than one creature. Thus is ineligible. It very clearly no longer "targets only you" or "targets only one creature".