A quick search on the LL hardware database gave 4 reports of Lenovo 3000 G410 running LL so there should be a reasonable chance of getting something working.
As the laptop is without OS, the first thing is to get an OS running from live media. Due to the failed attempts, the state of the hard disk should be reviewed using tools on live media and any problems tidied up before attempting to install any OS on it.
To try LL installation again, start by downloading the iso again.
Make sure the download is ok by sum checking the file.
Check the Lenovo BIOS settings related to booting and how secureboot is set (if applicable).
From the live boot review the state of the hard disk and sort it out if necessary.
Follow the OS install instructions carefully. Personally i don't bother with updates at the installation stage.
Making a note of each choice made and record any messages that result.
If problems occur don't continue but seek advice.
LL 5.x would be best, but due to the age of the laptop it might be worth looking at LL v4.x which is supported for a couple of years.
