Am I able to use my Android smartphone with 4G LTE internet access as some kind of gateway to internet?
I need anyone who is connected to my local network made with Wi-fi router to be able to access internet.
My Android phone is also connected to this network by Wi-fi.
I do not mean Wi-fi HotSpot on phone or WDS. Those are useless to me.
My phone is rooted so I am able to make some routing there if nessesary.
Is it even possible?
It is possible, though your Asus router may not support it.
Your picture is actually close to what you need. One correction is you need two wifi networks:
Then you need to configure your router to use "hotspot" network as gateway for all devices from "client" network. It'll probably involve NAT in IPv4 and potentially some tricky subnetting or address translation in IPv6.
Update: here's the picture: 
If the stipulation that the phone must be connected via wifi, or you don't have an already running windows machine, this answer wont be helpful to you, however;
If you use a USB cable and connect it to a windows machine, like a laptop (with an ethernet port) you don't use or something, hook it up and in the same place you usually select "share device files" in android instead select "tether internet".
Now in windows open up Network Connections (hit start and run ncpa.cpl) and right-click properties for your phone's provided internet connexion. Under the Sharing tab, hit "Allow...users to connect", and in the dropdown select the ethernet port.
Connect your ethernet to your router's WAN socket, and in your router ask it to use Ethernet WAN (disable VLAN) with DHCP routing (autodetect everything and enable NAT).
Your router will act as a gateway as normal, thinking your laptop is a dedicated WAN device like with some fibre-optic solutions use. All routers should be capable of this, don't forget to tell the machine not to fall asleep.
A proposal: You can use your phone connected through usb cable (usb tethering) to a server acting as router, firewall, even proxy. The usb connection will make a new usb ethernet interface, Internet facing, another interface, NIC or WLAN can connect to Asus wifi and provide access to the Internet to all your WiFi network.
Else, as Michael Hampton mentioned, you just do the same thing but with added complexity. Perhaps you need many concurrent connections, that's a scenario I can think of where there's a point connecting to a dedicated device for the task, than to mobile phone AP.
Sure, a phone as a hotspot is technically a gateway, but it's a super limited one. For instance, I cannot stream to my chromecast using my phone and phone as a hotspot because chromecast cannot connect to my hotspot network and I can't modify any of the network settings beyond the password.
I believe the solution would be to get an access point and then connect that to a router. You should be able to wirelessly receive and relay the phone data connection through ethernet on to the router.
Then it aught to be the same as if you're plugging in a modem, and you have complete control over this new network and can do MAC address filtering, port forwarding, network analysis, and all that fun stuff.
It seems nobody could answer this question. It's easy.
The phone is a gateway to the internet, and it needs to feed into the router's WAN port. The diagram in the first answer shows Router connect to Hotspot Wifi, but it doesn't say how.
You can't 'plug' a wireless connection into your WAN port. So you need a wireless bridge as follows:
4G hotspot WIFI WAN
[Internet]---------[Phone]>>> <<<[Bridge]---[ROUTER]----- client(s)
The "bridge" can be a 'travel router' a simple router that acts as a client to your phone's hotspot, and feeds its signal into the router's WAN port.
And you don't need a bridge, if you use USB tethering.
a) plug your phone into the router's USB port (super-speed USB3)
b) enable "usb tethering" on the phone
4G USB
[Internet]---------[Phone]---------------[ROUTER]----- client(s)
Now your router will get internet directly from the phone without a wireless hotspot. But both will work.
