Firstly, yes, if a spell requires an attack, that attack still counts as an attack.
So, what you have to remember is the "specific beats general" rule. This is detailed on page 7 of the PHB. The core of the rule is:
If a specific rule contradicts a general rule, the specific rule wins.
So the general rule here is that familiars can't attack. There are 2 exceptions to this general rule:
A familiar can deliver spells with a range of touch on your behalf, even if the spell requires an attack roll.
A Warlock with the Pact of the Chain can forgo one of their attacks to let their familiar make one.
So when you cast a spell with a range of touch, you can have your familiar do the actual "touching" on your behalf. You cast the spell as normal using whatever actions it requires. The familiar is required to use their reaction, and if the spell requires an attack roll, it makes the attack roll. Since it uses your modifier anyway, it's exactly the same as you making the attack roll except that you don't have to stand next to what you're casting the spell on.
In the case of a Warlock with the Pact of the Chain, the Warlock takes the Attack action as usual, then has the familiar do the actual attack. The errata for the PHB says:
When you
let your familiar attack, it does so with its
reaction.
If the Warlock can make multiple attacks with the Attack action, the familiar can use its reaction to replace one of them with its own attack, then the Warlock makes the rest.
Finally, only spells with a range of touch can be delivered by your familiar. There is no option for having your familiar cast a spell on your behalf, and the option for allowing a familiar to attack instead of yourself specifically says "when you take the Attack action", not just "when you make an attack".
Your familiar understands you perfectly.
You've already quoted nearly all relevant text in your answer. Telepathy (within range) communicates without necessarily using words, but the target (with int >= 1) understands perfectly.
I think your confusion comes from when you state
simple animal familiar gained
There is simply no such thing. From the spell description:
You gain the service of a familiar, a spirit that takes an animal form
and thus it's impossible for it to actually be a simple animal.
Best Answer
Yes, depending on how the spell targets creatures
Rubiksmoose gave an excellent break down of what an attack is here, but what's relevant to this question can be found on p.193-194, PHB:
As a familiar is not able to attack (excluding a Warlock Pact of the Chain familiar), but there is no mention to a familiar's ability to cast a spell, then the familiar should be able to cast any spell that does not require an attack roll. The other question that may come up is the familiars ability to complete the components of the spell, but as answered here in regards to magic items casting spells, DMG p.141 states:
The bead created from Arcane Abeyance is a magical item, and would follow the rules of magic items from above, meaning the components of the spell would be irrelevant to the familiar's ability to cast the spell. In regards to the concentration of the spell in question, Arcane Abeyance states, as you've linked:
The only places the spell would not use the familiars abilities are on the spell attack bonus (irrelevant for a familiar) or spell save DC. All other aspects of the spell would be the responsibility of the familiar, including concentration.