As I plan to sell this solution as a whole, do I have to publish sources of my Linux app?
No, if I assume that this is a typical userspace app, then the terms of the Kernel GPL license would not apply (though I still may combine my userspace app with copyleft-licensed otherwise and a copyleft may apply then).
From this other answer of mine:
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.
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.
or kernel drivers I have made?
Yes. if some of 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.
or deliver it upon request to the customer?
Yes, I have several options to deliver source code. One of them is to deliver source code on request.
Do I even have to make my UI-app and drivers under any GPL license?
See above: it depends. Typically the GPL would not apply to a common userspace app just based on it being a userspace app. The LKM/drivers should always be treated as GPL-licensed.
Do I need to tell the customer what Linux distribution I am using for this my solution, as he will never be able to interact with Linux itself, or use any of its functionalities directly (running terminal, or any other app)?
I need to disclose and attribute. Attribution is important in general (and license notification is an essential part in the GPL in particular).
The GPL always requires attribution composed at the minimum of a copyright statement, a notice and the GPL license text. Attribution examples are provided at the bottom of the license text itself.
See also these answer of mine for more details.
What freedoms I have to change this Linux distribution (remove parts that I don't need for current use)?
I have full freedom to change an Linux distribution (assuming this is a true open source distribution) including adding, changing or removing. Yet, in doing so I may have extra obligations. A typical distro contains 1000+ packages beside the Kernel and each may have a different license and specific obligations. Some licenses require source redistribution, some do not. Most require some form of attribution. Some require change tracking. A few may prohibit certain changes. Some distros may also include packages that are proprietary and/or not redistributable directly such as Ubuntu and RedHat branding.