Dog runRace

I suppose you mean "dog race", there is no such thing as a "dog run".

I took my friend's dogs to thea dog runace today.

These names comes from milktea as you might thinkhave thought, because my friend really likes milktea.

They are lively and like to play outside, so they looked really funit was fun to watch them.

In fact, I carried a dog for the first time today, so I was very nervous for due to fear that I might drop the dog.

And I thoughI think that spending time with dogs can heal the mind.

Dog runPark

I assume you mean the word ドッグラン (dog run). We call them dog parks. :-)

I took my friend's dogs to the dog runpark today.

These names comes from milk tea as you might think, because my friend really like milktea.have guessed. My friend really likes milk tea.¶
(Alternative: My friend named her dogs after milk tea because she really likes it as you might have guessed.)

They are lively and like to play outside, so they looked reallylike they were really having fun.

In fact, I carried a dog for the first time today, so I was very nervous for fearafraid that I mightwould drop dogit.

And I thoughtink(?) spending time with dogs can heal the mind.

If you thought it, do you no longer believe that spending time with dogs can heal the mind. It sounds like you want to say here: "And I thought spending time with dogs can heal the mind, but I was wrong."

Dog runRace

I've never heard of "dog run" before, only "dog race" or "dog show"

Dog race -> where the dogs run
Dog show -> usually a beauty or talent show with the dogs

I took my friend's dogs to the dog runace today.

These names comes from milktea as you might thinkhave thought, because my friend really likes milktea.

They are lively and they like to play outside, so they looked really fun.it was really fun to watch them.

"They looked really fun" sounds awkward, I don't think someone can "look fun"
You can say as an alternative....
".. so it was really fun to watch them."
"... so they looked like they were having fun."

In fact, I carried a dog for the first time today, so I was very nervous forwith the fear that I might drop the dog.

Andlso, I thought spending time with dogs can heal the mind.