A few of these questions overlap, but I'll try to address them individually.
Can an individual nanoprobe act intelligently?
Its implied Yes.
TUVOK: No, but we must remember that she's experienced hallucinatory images before.
EMH: That was in direct response to a signal from the ship where she was assimilated. She's not having hallucinations now. She's remembering what happened to her. I've confirmed this by analysing the specific engrammatic activity in her hippocampus. We're not talking about conjecture, we're talking about science.
JANEWAY: Let's not get bogged down. Seven's made serious accusations and I won't dismiss them. If Kovin assaulted her, took a sample of her Borg technology, we can't stand by and do nothing. In the wrong hands a single nanoprobe could lead to disaster. Doctor, I want you to keep searching for any physical evidence to back up Seven's claim. I will talk to Kovin. Dismissed.
ARCHER: Any progress finding a treatment?
PHLOX: Possibly. I've been attempting to treat the infection as if it were, well, an infection. Then it suddenly occurred to me, this isn't a biological problem. I need to think like an engineer, not a physician. I extracted several of the devices and subjected them to different forms of radiation. Their intra-molecular processors appear to be vulnerable to omicron particles. But if one of the nanoprobes survives, it will start to multiply again. So I'll need to programme a rather excessive dose. The side effects will not be pleasant, to say the least. You may want to keep this in the event I'm unsuccessful.
These examples could be interpreted as a single one can replicate, but to be useful a Nanoprobe would have to replicate and then program its clone, otherwise every probe would be identical, (an army of nanoprobes all doing the same job wouldn't be helpful. E.G. If they all build an eyepiece which nanites are going to build the armor). To replicate, then reprogram the clone to do a different task would require intelligence and communication. Which brings us to the next question.
Can a collection of nanoprobes (say in the same host) communicate with each other and act intelligently en masse ?
Yes, this would be necessary to assimilate an individual, first they assimilate the blood, but then would have to coordinate to build all the implants, each building a different component. In the episode Voy:Drone the nanoprobes infect the mobile emitter, and create a new Borg using a genetic sample: this isn't the Borgs usual reproductive method. To design a new drone from scratch (without any help from the collective), would require a high level of intelligence and communication between them.
Do nanoprobes respond to orders from their host?
Again this has to be Yes.
7 of 9 uses the nanites to try to disable a warhead Voy:Warhead, transfer the Doctor to and from her implants Voy:Body and Soul , interface with voyagers helm control Voy:Scorpion Part2 ect. Borg use them to create new Drones and grow Borg tech after injection too. To do all these different thing you would have to tell the nanoprobes what to do before injection. Otherwise they would just assimilate whatever they were injected into, this strongly indicates the nanites are programmed by the hosts mind.
Do nanoprobes respond to orders from the Borg collective?
Possibly, but I suspect this is a no
(Speculation). Given the reason in the last question its logical to assume they take instructions from the host. The Borg have never been seen giving remote commands to the nanites so its likely the collective gives the Drone instructions, then the Drone instructs the nanites. There would be no need to update the nanites directly from the collective. Any updates can be given to the Drone to then give to the nanites.