"In my friend's house, there are three cute dogs."
This is called phonological assimilation - a consonant becomes more like the previous one. In this case, "h" assimilates to sound more like "n" that precedes it.
It's very common in Japanese when "n" and an unvoiced consonant meet (usually "h" or "k"). H- syllables then become their voiced variants, b- / p-. Similarly "k" becomes "g". Contraction and assimilation of いち and ろく when combined with counting words is also common.
Some examples: さん+ひき --> さんびき (three small animals)
いち+ひき --> いっぴき (one small animal)
さん+かい --> さんがい (third floor) BUT さん+かい --> さんかい (three times)
さん+ほん --> さんぼん (three long cylindrical things)
If you don't know the correct variant from memory, it's safe to assume that if there's いち or さん before the counting word, and the counting word starts with unvoiced consonant, it will assimilate.
Is there any reason that には in this sentence shouldn't be translated to "at" instead of "in?" The English sentence implies that the dogs are inside the house, while the Japanese sentence seems to imply that the dogs are a part of the household. I would never interpret the English sentence as three dogs are part of the friend's household. With that wording, I would think the friend is possibly dogsitting or fostering some/all of the dogs. Is there some nuance in the Japanese phrase "友達の家には" that I'm not picking up on or does it really imply that there are three dogs inside the house?