Any way to create a Druid subclass where they can add a Sorcerer spell (their choice) to their class list at level 1, 2, 4, and 6, and add a Bard spell to their class list at level 1, 3, 5?

Yes. Please see the Homebrew Subclasses & Species FAQ, question #15: (https://www.dndbeyond.com/forums/dungeons-dragons-discussion/homebrew-house-rules/131411-a-homebrewers-how-to-faq#HomebrewSubclasses&SpeciesFAQ).

Thank you.

1. Because I'm adding these to a Druid, do they just become available to the character as spells they can prepare as a Druid, or do they become permanently know, or is that controlled by some combination of the "Consumes Spell Slot", "Counts as Known Spell", and "Always Prepared" options?
2. Here I'm working toward a Druid, but are there "gotchas" that I should look out for depending on whether spell selection is like preparing spells for {Druid, Cleric} (or Artificer at 1/2 levels), or learning spells like {Bard, Sorcerer, Warlock} (or the 1/2 or 1/3 casters), or that Thing Which is Wizard?
3. It looks like I have to add individual spells one at a time rather than saying "Give them one Bard spell at each of levels 1, 3, 5"? Is that correct? A bit awkward, but definitely workable.

Quote from Qzot >>

Thank you.

1. Because I'm adding these to a Druid, do they just become available to the character as spells they can prepare as a Druid, or do they become permanently know, or is that controlled by some combination of the "Consumes Spell Slot", "Counts as Known Spell", and "Always Prepared" options?
2. Here I'm working toward a Druid, but are there "gotchas" that I should look out for depending on whether spell selection is like preparing spells for {Druid, Cleric} (or Artificer at 1/2 levels), or learning spells like {Bard, Sorcerer, Warlock} (or the 1/2 or 1/3 casters), or that Thing Which is Wizard?
3. It looks like I have to add individual spells one at a time rather than saying "Give them one Bard spell at each of levels 1, 3, 5"? Is that correct? A bit awkward, but definitely workable.

1.  It depends on the combination of settings you choose, please see below.👇
2. Yes, please see below.👇
3. You don’t have to add each individual spell option as a choice, but you do need to add each instance of choice individually. I’ll explain a bit more further down. ⬇️

👉 The thing to remember is that some classes automatically “know” all of their spells and get to “prepare” the ones they want, other classes have to “learn” individually spells but can cast any that they “know,” and wizards have to learn their spells individually but can learn a whackton and then prepare the ones they want. In DDB’s system there is a setting for each base class (that we as homebrewers cannot access) labeled “knows all spells.” If that setting is toggled on like it is for artificers, clerics, druids, and paladins, then the “always prepared” setting you mentioned becomes relevant and the “counts as known” setting you mentioned becomes irrelevant. If that setting is instead toggled off as is the case for bards, rangers, sorcerers and warlocks (and wizards), then the “always prepared” setting is irrelevant but the “counts as known” setting becomes very relevant (except for wizards).

The “always prepared” setting is obviously irrelevant if PC‘s with that class don’t prepare any spells. Wizards are a weird exception in that they do prepare spells, they just instead get treated as if they don’t because they don’t automatically “know all spells” on their class list and still have to “learn” them. The “counts as known” setting is either relevant or irrelevant depending on if PCs of that class only get to learn a finite number of spells. If you select “counts as known” for a spell attached to a druid subclass it is simply ignored by the system because druids have no limit on the number of spells they can learn. Conversely, selecting that for a spell attached to a sorcerer subclass would eat into the number of spells that a PC using that subclass could potentially learn. Wizards again ignore this setting too as they also have absolutely no limitations whatsoever on the number of spells they can potentially learn.

So, if you attach a spell to a druid subclass you can completely ignore the “counts as known” setting as it is irrelevant. If you want them to always have the spell prepped at all times then you definitely want to say “yes” to “always prepared.” Normally a subclass that gets spells automatically granted as “always prepared” like an artificer, cleric, druid or paladin would not have those spells added in this way at all. For those classes, those always prepared subclass granted spells are usually just entered in the “additional specific spells” field under Basic Information for that subclass, and the system automatically handles granting them at the correct levels and keeping them always prepared. The reason you can’t do it that way is that you instead want to present the player with choices, so you have to do it the long way,

The only “gotcha” is that if you enter spells into the “additional specific spells” field under Basic Information for a subclass for any class that has to ”learn” their spells (including wizards), then those spells are not automatically known but instead are simply added to the list of spells the character could learn the way it works for warlocks and their patron spells. If you instead want those spells automatically known by those PCs, then you have to add them under the Spells tab like you are doing for your subclass.

➡️ You have to tell the system “give them any one 1st-level bard spell of their choice at level 1,” and then separately tell the system “give them any one 2nd-level bard spell of their choice at level 3,” etc. because you have to specifically designate which spell level the get to choose from at each level. That’s why. Otherwise it would let them choose 9th-level spells at level 1.

You can learn more about all of this stuff simply be reading questions 13-15 in that same Subclasses & Species FAQ I linked for you in my previous post.