I would like to know if selling a hardware product which runs a firmware is considered as conveying that firmware.
Yes. The binaries are inside the device and are therefore redistributed with the device aka. conveyed.
For example, if I want to use a GPLv3 library statically linked with my own code, I am required to inform the customers that buy my products that they are running GPLv3 software?
Yes, notification is the essence of most open source licenses including the GPL.
And licensing also my code as GPLv3?
Yes, when linking code (statically or dynamically) with GPL-licensed, the copyleft of the GPL would typically apply to my code and should be made available under the GPL or a GPL-compatible license and treated as GPL-licensed. You can also find more details on dependencies in this answer:
For Copyleft licenses, how (proprietary- or non-Copyleft- licensed) programs and Copyleft-licensed programs are used together, how they depend and interact with each other is the essence of what triggers the Copyleft clauses of the GPL and LGPL.
In the case of the GPL, I consider any linking as an "intimate" relationship where the copyleft would apply and extend to the caller.
In such case, how I must inform them?
This can take several forms such as notifying in the UI (if my device has such a thing) or the documentation. It should contain at least a notice and the text of the GPL. See also this answer (of mine) on: Does GPLv3 require attribution?
And about a different but similar case: if I decide to employ uCLinux (a linux version for embedded products), can I ship the product with close source programs, which will run over the uCLinux OS, and offering the source code of uCLinux only?
Yes, if my closed source code runs in userspace, this is considered a normal usage of the OS and I can use any license I please.
Linux (the Kernel) uses the GPL 2.0 with an extra statement per this answer:
NOTE! This copyright does *not* cover user programs that use kernel
services by normal system calls - this is merely considered normal use
of the kernel, and does *not* fall under the heading of "derived work".
Also note that the GPL below is copyrighted by the Free Software
Foundation, but the instance of code that it refers to (the Linux
kernel) is copyrighted by me and others who actually wrote it.
No, if my closed source code runs in kernelspace, then the consensus is that this code should be licensed under a GPL or GPL-compatible license and treated as GPL-licensed.
And of course my device would typically combine both userspace and kernelspace code so a bit of both may apply.