Entering the house, I tripped over the carpet.
Reading what the RAE has to say, there seems to a distinction between tropezar con un objeto (to stumble on something) and tropezarse con una persona (to run into somebody). That said, it seems that most sentences in the corpus use the reflexive version for both purposes.
Yeah, you’d usually use tropezarse with people and tropezar with things.
That said, I think there is a tendency to use the reflexive even when is not necesserly needed. So that part of the sentence sounds kinda of ok.
Personally have more issue with “entrar a la casa” (I actually thought that’s what the post was going to be about, before opening it). It sounds off. I’d suggest “en la casa” or, if its your home, “en casa”.
I’ve seen it often enough to almost incorporate it in my own patterns of speech, so I’m glad you broached the subject. Google Ngrams confirms that en is the more frequent preposition:
entrar a la casa 0.0000409715%
entrar en la casa 0.0001097966%
entrar a casa 0.0000069539%
entrar en casa 0.0000658387%
Since it appears in Ngrams, I wouldn’t rule it out completely either. But maybe its a regional thing or an influence of another language (in catalan, for instance, you do say “entrar a casa”).