1. The lyre should be treated as slotted.
Given that you require the lyre to be held in both hands to be able to use it's magic abilities, you should treat it as a slotted item, as it takes up both weapon slots.
The rules don't say so, but you must hold a magic weapon in one or two hands (as appropriate for the weapon) to use any of its magical properties. You cannot, for example, use a sword's spell-like abilities while you have the sword put away in its scabbard.
(quoted from here)
This gives precedence that, while not stated in the core rules there are two weapon slots. Given that the lyre requires use of both your hands and prevents you from using a magic weapon (or any weapon for that matter) at the same time, it would make sense that the lyre makes use of both of your weapon slots and is thus a slotted item.
2. Yes, your calculations are correct
Any one of the similar abilities on its own item would make up the base cost for that item, so having multiple of them on one item means that they all contribute to the base cost, while the No Space Limitation states that it modifies the entire cost of the weapon by two. So:
& 2 \times (100\% + 75\% + 50\% + \ldots)
=\ & 2 \times 100\% + 2 \times 75\% + 2 \times 50\% + \ldots
=\ & 200\% + 150\% + 100\% + \ldots
for the final cost of multiple similar abilities.
3. No, similar never costs extra, and it depends
First, let's cover "Do “similar” abilities never cost extra?"
From Multiple Similar Abilities we know that similar abilities cost less on unslotted items. And from Multiple Different Abilities:
Abilities such as an attack roll bonus or saving throw bonus and a spell-like function are not similar, and their values are simply added together to determine the cost. For items that do take up a space on a character’s body each additional power not only has no discount but instead has a 50% increase in price.
we can see that different abilities are just added on at base cost on unslotted items. So, for unslotted items, similar abilities will never individually cost more than their base cost.
However, for slotted items the idea of "similar abilities" doesn't seem to exist as Multiple Similar Abilities is explicitly stated as only for items "that don’t take up space on a character’s body", and Multiple Different Abilities states:
For items that do take up a space on a character’s body each additional power not only has no discount but instead has a 50% increase in price.
Together, these two entries imply that slotted items never get discounts to their individual abilities, similar or not.
Answering what is a similar ability and what is not is a bit of a tricky question. The short answer is "it's up to the DM to decide" as we aren't given a strong definition.
If we go back to Multiple Different Abilities:
Abilities such as an attack roll bonus or saving throw bonus and a spell-like function are not similar
A little basic logic allows us to determine that clearly attack roll bonuses and Spell Effects are not similar abilities and neither are saving throw bonuses and Spell Effects. We cannot, however conclusively determine from this text if attack roll bonuses and saving throw bonuses are supposed to be similar or different abilities. Though it could be taken as implied to be similar as they are contrasted as different to spell-like features together.
Furthermore it doesn't give any indication whether or not all spell-like features are considered similar or if they are all considered different to each other, or if it is dependent upon the spell used in the features themselves.
So, ultimately, similarity is defined by the DM's interpretation (or the player's with DM approval) when creating magic items.
4. This is a general rule
After digging out my DMG to find which examples you were referring to, and running the numbers for said examples, it appears to be a general rule that the gold values for the Intelligent Item abilities are just a flat increase in gold cost, they don't count as similar abilities.
This was determined by calculating the cost for several of the example items and always coming close to or reaching exactly, the listed price for the items. In contrast, if each of the Intelligent abilities were added to the price, but also using the Multiple Different Abilities rule (as discussed for slotted items in section 3) then the results would be far greater than that of the listed prices in the book.
5. I can't speak about a consesus, but...
I can tell you what I believe the reasoning is behind the discount in my (very limited) game design experience.
The reason the skill requirement is a 10% discount is because it is an easy requirement to overcome. Initially, a skill requirement means that the item is only useful to those who would take ranks in the required skill regardless, however, anyone who wanted to use it badly enough could burn a couple skill points in the required skill to use the item.
While a class/alignment requirement to lower price may seem like cheese for building more powerful weapons for the same price (or equally powerful weapons for less), it is because the requirement is quite a bit more limiting than just a skill.
An alignment requirement means that if you are not of the matching alignment you have to change it, whether through magic or acts. You have to change your characters fundamental beliefs/attitudes/behaviours/etc. just to use the item if you aren't already matching it. A class requirement means that, if you want to use the item, you need to take a multiclass dip into a class that you may not particularly want to.
While, yes, a class/alignment requirement is a bit of a cheese way to lower your item cost when making an item for yourself, it is a genuine restriction when making magic items without a particular party or player in mind.
Final Question: Yes, you can make an item more durable
While I don't know of any methods off the top of my head to make a lyre more durable in-game, I figure that it is fine to do so as part of its creation. All items have an AC, Hardness, and HP. All of these can be set to whatever value you wish to make them more durable, but you will need to okay it with the DM. More than likely you will need to come up with some justification like it being made out of a special material or being ambiguously, magically reinforced, and doing so may incur greater cost.
AC for an item works just like AC for a character, but is generally low as items have Dex 0 (-5 mod), and then just a bonus -2 again for being an item. There is also no precedent for increasing AC with special materials in the core rules, so increased AC would probably be a hard to justify thing.
HP for an item also works just like character HP. Anything more than 0, and the object is good to go. Thus, the more hit points that an object has the more of a beating it can take. However, restoring these hit points is done through an appropriate Craft skill, so if you don't want to or don't have the spare skill points to invest, then this will probably be easier to justify, but only lasts as long as your object has HP.
Hardness for an item works like DR/- for a character. It is just a flat reduction of damage dealt to the object with no bypassing it. This will probably be the best option, as it will be easier to justify and will do as good, if not better, job than a either buffing AC or HP.