Hey,
Should someone have same problems as I described:
1) APN was not correct, when fixed, I was able to access internet
2) I had to request public IP from mobile network provider. It was not possible to access the device remotely otherwise
3) after I set port forwarding, it started to work, but I had to redirect port 80 to other one as it is used by synology
All is good now. Thanks for the help