Strictly better? No.

One of the major points of Disguise Self is that it can not only alter your appearance (via illusion), but your clothing and equipment as well.

It's important to note the inverse of this: Alter Self does not specify that it modifies your clothing or equipment. This means that, depending on how severe your alterations are, you may no longer fit into your armor and your clothing may clearly no longer fit you, depending on how simulationist your DM is on the topic. This is especially important if using the "Variant: Equipment Sizes" rule on PHB p.144:

In most campaigns, you can use or wear any equipment that you find on your adventures, within the bounds of common sense. For example, a burly half-orc won’t fit in a halfling’s leather armor, and a gnome would be swallowed up in a cloud giant’s elegant robe.

The DM can impose more realism. For example, a suit of plate armor made for one human might not fit another one without significant alterations, and a guard’s uniform might be visibly ill-fitting when an adventurer tries to wear it as a disguise.

Using this variant, when adventurers find armor, clothing, and similar items that are made to be worn, they might need to visit an armorsmith, tailor, leatherworker, or similar expert to make the item wearable. The cost for such work varies from 10 to 40 percent of the market price of the item. The DM can either roll 1d4 × 10 or determine the increase in cost based on the extent of the alterations required.

Even if your equipment does fit you after the effects of the spell, any well known gear or accessories may still let you be identified, especially if they're more well known then your actual physical attributes.

There are a few other considerations as well.

One of them is that both Alter Self and Disguise Self can be used at the same time, to enhance the illusion with actual physical changes underneath it. Do note that this is only possible because Disguise Self does not require concentration, which is another point in its favor over Alter Self, which does. This means Alter Self can fail earlier than intended (especially in combat or while under stress), while Disguise Self will last the duration under most conditions, and still allows you to cast other concentration spells.

Another is comparing how Disguise Self and Alter Self interact with height & size.

Alter Self actually changes your height, while Disguise Self only creates the illusion of a height change, keeping you your usual size. This might matter if your DM actually cares about character height in certain situations rather than just your size.

In addition, Alter Self restricts you to only changing your appearance to a creature of the same size with no other restriction on maximum height gain/loss, while Disguise Self only has a restriction on height change rather than size change.

For example, with Alter Self, you could change the shortest possible dwarf into the tallest possible goliath, whereas with Disguise Self, you could make a dwarf appear to be a halfling (so long as you're not trying to change your height by more than a foot in doing so).

Overall, I'd probably rate Alter Self better than Disguise Self in some combination of the following situations:

• There is plenty of time to make preparations in advance, including having an appropriate alternate outfit ready.
• You are unprepared, but you only need minor superficial changes with no drastic change such as size, making the lack of alternate costume not an issue.
• You are in no danger of having your concentration broken for the next hour, nor are you likely to need to cast any other Concentration spells.
• You are likely to be thoroughly inspected for some reason or another.

In other situations, there is a chance Disguise Self may be the more appropriate spell. In a situation requiring an emergency drastic appearance change (clothes, size, and all), for example, I would much prefer Disguise Self as my option.

No, you can't cast it using a spell slot.

Spells you can cast because of your race aren't spells you know, and thus can't be spells you prepare. Per the PHB (p. 201, "Known and Prepared Spells"):

Before a spellcaster can use a spell, he or she must have the spell firmly fixed in mind ...

... and it goes on to describe some of the class-specific variations of knowing and preparing spells. But in any case, having a feature from your race or class that allows you to cast a spell isn't the same as knowing the spell.

This is spelled out clearly in the multiclassing rules for Pact Magic quoted in the question (PHB p. 164; emphasis mine).

If you have both the Spellcasting class feature and the Pact Magic class feature from the warlock class, you can use the spell slots you gain from the Pact Magic feature to cast spells you know or have prepared from classes with the Spellcasting class feature, and you can use the spell slots you gain from the Spellcasting class feature to cast warlock spells you know.

Likewise, the Drow Magic trait clearly distinguishes the known cantrip from the merely castable other spells (PHB, p. 24; bold for emphasis mine):

You know the dancing lights cantrip. When you reach 3rd level, you can cast the faerie fire spell once per day. When you reach 5th level, you can also cast the darkness spell once per day. Charisma is your spellcasting ability for these spells.

If the intent of the rules was that Drow also knew the non-cantrip spells, it would say so.