Hi

1. I agree that BUT seems an unusual conjunction in this case but its 100% accurate, however, I would suggest adding "still" in order to stress that there is a deliberate difference in the conjunctive clause - "but I still like my Orange pillow very much" - its an unusual sentence structure but natural for younger people (slight stress on the Orange here)

2. Weary is perfectly fine here, we all feel weary, however, homework here is a pronoun, suggest "it" rather than "them"

3. In common usage, we don't use "self-introduction", we just conjugate and use "her introduction", "my introduction", "his introduction", "your introduction", etc -- imo, in common usage, to say that someone had a "self-introduction" would imply that they were belittling the recipient and trying to be more grandoise than they should be.

We Yanks might also say 'I'm just an ordinary guy/girl' and it doesn't sound negative to my ear.  'I'm an ordinary girl', 'I'm a very ordinary girl'... they both sound humble, but not necessarily bad.  Maybe it's a Yank thing?  Maybe it's a me thing.

I agree with you guys about the oddness of using 'but' in this situation.  The above two alternatives sound much better.

I agree with Kieran.  I've seen "self-introduction" used a lot recently but only by non-native speakers.  

You could change "but" for "though". It would be my preference.