“Provide us a great hospitality” can be either, “make us feel welcome” or “be a hospitable” or “be a good host.”

I’m not sure if this is what you mean. The way you wrote it means that your friend had a hard time making you feel welcome … in other words, the friend was not a good host. However, you might have meant that your friend went to a great effort to make you feel welcome. So you can say:

“My friend was hard-pressed to make us feel welcome” or “My friend was hard-pressed to be a good host” means “My friend didn’t do a very good job at hosting us” OR:
“My friend worked hard to make us feel welcome” means “My friend worked hard to make us feel welcome (and we felt welcome)”

Perfect examples. “Due” doesn’t sound like the right word here. You might say, “hard-pressed to provide adequate medical care.”

You need to say, “hard-pressed to…” plus an action. So you can say, “The coach was hard-pressed to answer the journalists’ questions in the post-game interview.” This means the coach had difficulty answering the questions (the action is “to answer”). However, you can say, “The media pressed the coach hard in the post-game interview.” That means something different – that means the media asked him hard questions (and didn’t let him avoid the questions).