The SD card slot of my Raspberry Pi was broken. I tried to replace (solder) it with one that i bought at eBay but with no success.
I know that a SD slot is required but is there another a way to get my Raspi working without a SD card?
No, you absolutely positively can not boot without SD slot.
You should get another Pi or try to fix the one you have.
My Raspberry Pi 3 is booting without an SD card.
I flashed the same distro image (LibreElec) onto an SD-card and an USB stick (using Linux and with the
dd command). After that I've changed the
cmdline.txt in the boot partition from both, SD-card and USB stick.
Finally I've added
program_usb_boot_mode=1 to both
config.txt files, which enables booting from usb stick.
I don't think that'll help you, however it could be helpful for someone else.
Buy an SMD REWORK STATION and try put it out, clean the zone that being used and then try solder again as it should be, or you just can buy another RPi.
Here are instructions to fix the SD card slot.