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:
let your familiar attack, it does so with its
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".
It's almost correct, but not quite.
There are two separate, unrelated ways to get a familiar in D&D 5e, and you're trying to combine them.
The first method is by use of the Find Familiar spell, which allows you to summon a celestial, fey, or fiendish spirit that takes the form of any of a list of creatures. This list is expanded by the warlock pact of the chain. This familiar is perfectly obedient, can be resummoned when it dies, can be hid in a pocket dimension, deliver touch spells that you cast, and everything else specified in the spell description. This method gives you a familiar with the basic stats of the chosen creature, not the "variant: familiar" traits of the chosen creature (unless your DM chooses to have that creature type appear).
The second method is by finding a quasit, imp, or pseudodragon that has the "variant: familiar" trait (which is 100% up to the DM), and enlisting it as a familiar by interacting with it. This familiar has only the traits listed in the stat block for that creature, including the variant traits, but none of the traits of familiars given by the Find Familiar spell. No pocket dimension, no touch spells, no limitations on what actions it can perform, and if it dies, it's just dead. This type of familiar is an NPC controlled by the DM (much like a hireling or other follower), and is only as obedient to the PC as the DM says it is, using the MM entry as a guide.
Now that we've established how things actually work, we can address your real concern. Your warlock can't yet communicate at a great distance, but he can soon. There is a warlock invocation available to him called Voice of the Chain Master that does the same thing, but with unlimited range on the same plane. As you've realized, this ability has some incredible potential, especially for scouting.
It's not any more powerful than other options and should not be limited.
Let's compare it to some other invocations. There's one that lets a warlock cast Disguise Self as at will. This would let him see a guard, and appear exactly like that guard and just walk around the enemy camp unimpeded. Or maybe impersonate the leader of the camp and just take it over without even a struggle. There's another invocation that lets the warlock cast arcane eye at will, which gives you a way better scout than an easily killed creature. A familiar, even an invisible one, still has to succeed on a Dexterity (stealth) check to avoid being heard and then easily killed. An arcane eye does not.
So, in order to fully utilize this scouting ability, your warlock has to pick the chain pact and spend one of his few, precious invocations, both of which are huge opportunity costs. He deserves something in return. This something is you not limiting it. It already has a flaw in still being able to be killed by anything that hears it, or smells it. That's right, just about any pet wolf is going to be enough to catch this familiar. It doesn't need any more limitations.
Yes, it can act on its own.
First, let's cover some basics: since a familiar is a class feature, not an NPC, it should be completely under the player's control.
The primary source even states, and I quote, "... the familiar and master are one being. "
As such, the DM has no business taking over your familiar... unless that is a house rule your group has agreed upon.
As such, the actions of a familiar are supposed to be under the control of the character, and thus player, to dictate.
Next, as the familiar counts as a creature, it gets its own actions. It also has an initiative entry on the official character sheets for familiars, mounts, and cohorts, which can be different from it's master's initiative.
Since the character, and thus player, is or should be, in control of the familiar, they can instruct it to remain within 5', or move away for an action, with all the tactical consequences thereof.
Note that the higher initiative being can delay their action, thus changing their initiative count to match the slower one.
If you are working under a house rule where the DM controls the familiar, then request the familiar (DM) to synchronize movements with your character in order to take advantage of the ability.
Due to the increased intelligence and communication abilities of a familiar, you should be able to communicate this or even train together without too much fuss. There are also training mechanics and rules in source books that could represent this capability, if the DM insists.
Thus, you don't have to limit a move to only 5', so long as they both synchronize their actions, and possibly have one delay their action the first round of combat.
It is worth noting that it is a very commonly adopted house rule to have the familiar simply act on the master's turn.