Does 'do not disturb' stop incoming texts ?

I'm confused what you mean by "that phone." Have you put a phone into DND mode, then sent a message from it? In that case the message will be sent. Or do you mean someone sent YOU a message and your phone was in DND mode? In that case the message will be received, and be viewable in your Messages app, but you will not get an alert that it arrived on your phone. But you will see it when you open Messages, and it will display the number of messages received while in DND mode in a badge on the Messages icon.

If Do Not Disturb's Repeated Calls feature is turned on, then if someone calls the phone twice within three minutes, the call won't be silenced.

Use Do Not Disturb on your iPhone, iPad, or iPod touch - Apple Support

Yes, but you won't get alerts that they have arrived.

You misread his response. You will receive the text but you will not get a notification.

🙂If somebody has do not disturb on their phone will they read my messages

They will receive the messages but not an alert that they have been received. Next time they pick up the phone, the messages will be there. Whether he reads them or not, it will be up to them.

If this person has forgotten to turn off do not disturb can telecommunication company like Testra get through to him if it's an emergency

In general, no, any more than they could get through if the phone was powered off or out of range of the network. Government Alerts in the U.S. will get through, as will VIP calls and texts, and repeat calls. But all of those except the government alerts are options up to the user.