I don't really see how ejecting the disc has any actual effect on the game crashing. If the disc is in good enough condition to install the game, it's in good enough condition to help it pass the disc check.
You say you're not good at computers and you're not confident what graphics card you have so I was thinking that you should try a disk utility repair permissions...
Press Command + Spacebar to open up a spotlight search. Then type in the phrase disk utility and press enter. This opens up the Disk Utility app. The app is split into two panes, left and right.
On the left hand side, click on your Mac's hard drive partition, probably named Macintosh HD. Then in the right hand side window, below the big empty white box, click on the Repair Disk Permissions button. This will take a few minutes so enjoy your cup of coffee or tea.
When it's done, try running Rome TW again, but don't bother trying to eject the disk. Let us know how it goes.