Run ALFA AWUS1900 (rtl8814au)

is there a way to run the USB3-Wifi-Adapter ALFA AWUS1900 with rtl8814au-chip under openwrt?

"Run" perhaps (if you go through the pain to find the github repo of the day and package that for OpenWrt), but definitely not in AP mode.

can you advice me a better device with USB, extern antenna, stable with openwrt, low price

Sure, anything but USB (preferably PCIe from QCA or Mediatek).

I have used various chipsets including the RTL8814 and the RTL8192.
I am using a USB Wifi Adapter and have no issues at all.
If you need drivers you can get them from Realtek.

With the AWUS1900 That has four Antennas... make sure that you have the correct driver that supports 802.11 ac/abgn and 802.11ac 4x4, 3 spatial streams MIMO
Below I listed the correct driver. You need the Realtek RTL8814AU The cable that came with the unit I dont know the length of yours, Mine came with that same dual plug cable whatever its called 21 pin or something they came with Android phones one of the Galaxy models...anyway..., If you need it longer DO NOT USE A USB Extension. Buy another Cable
I have almost the same unit as you, I upgraded antennas...so I have 6dbi and I also have 9dbi antennas. This is overkill for most but it depends on signal. Download and install Vistumbler If you do not know your signal and strength you need to know that, it goes hand in hand with antennas

Get Vistumbler will show the Wireless Connection with the Wifi Adapter in Real Time.
You can use this to adjust where you have the antenna and position of the antennas.
https://www.vistumbler.net/ Download link is right on this page

Your AWUS1900 Speeds - If signal sucks switch Wireless modes in the Advanced Settings for the Adapter
802.11 b: up to 11Mbps
802.11 g: up to 54Mbps
802.11 n: up to 600 Mbps
802.11 ac: up to 1300Mbps

Realtek RTL8814AU
VERSION: 1030.25.0701.2017
Updated: September 7, 2018

For Realtek ** USB Family Controller** Not the Driver for a Card!
. The driver supports the following USB products: RTL8812AU/RTL8811AU/RTL8192EU/RTL8812BU/RTL8811CU/RTL8814AU
DOWNLOAD DRIVERS
RTL8814AU Software : https://www.realtek.com/en/component/zoo/category/rtl8814au-software

Realtek ALL OTHER NETWORK ICs SOFTWARE From RTL8819-RTL8821
Use this Link https://www.realtek.com/en/component/zoo/advanced-search/514?Itemid=276

A huge issue I had with this chipset is the advanced settings.
Make sure that you configure them properly.
I had to change a few settings to get them to work properly.
You may want to try either 2.4 or 5ghz instead of Auto
There are a crap load of settings This is what I have set from default, you may be different.
After you change 1 thing. Hit ok and test it out.

These were old settings that worked for me
BeamCap, Disabled
AMSDU AMPDU, In Mode 1
LdpcCap - Ldpc Disabled
StbcCap - VHT STBC TX| HT STBC RX|TX
USB SF MODE - DISABLED
VHT 2.4G - Enabled if on b/g/ n
Wireless Mode a/n/ac Try b/g/n for 2.4g

Did you notice that this forum is about OpenWRT which is Linux distribution?
Everything what you wrote, and links you posted - all these is not related to Linux anyhow.