Is this miss as in the dog longs for the toilet? Or does the dog have bad aim?
The dog longs for the toilet. If the dog had bad aim the sentence would be "הכלב שלי מחטיא את השירותים" (max'ti)
Would it be correct to say that you use ל after the verb to indicate missing an object, and you use אל after the verb to say you miss a person?
No. Both can be used in both cases. Mostly use you would ל for everything, not because it's more correct, but because אל has fallen out of favor. However, there are a few special cases where only אל is correct. I can only think of the pronouns: מתגעגע אליך, מתגעגע אליו.
Side note: As far as I know, it is also correct to use מתגעגע על־, but have never seen this used in real Hebrew, either in writing or in speech.
"My dog needs the bathroom." Why was this marked wrong?
I don't really get this sentence. Is it that the dog yearns for the bathroom because it's a place he hasn't been in a long time. Like the dog yearns to go to the park. Or the dog yearns for the food on the table.
Or is the intention of the sentence that the dog needs to pee or poo?
אני אתגעגע אותך בגעגועים גדולים יותר מקטנים. I will miss you with longing bigger than little ones