I missed your post in the introduction forum, welcome to the site. I'm sorry to say that I am unable to identify your tree from that picture and without an ID it is difficult to offer specific advice. Perhaps if you post a new picture of a leaf it will help. Maybe someone else can ID it.
I usually suggest that all trees be outside during the summer but considering the cool nights we are still experiencing that may not be prudent for a few finicky species just yet. All of my plants are outside now, even Jades and Ficus, so the odds are that you will be OK.
The soil appears to be rather dense and peaty. Be careful not to water too frequently as this tree appears to be over-potted and in distress. All of which can lead to a situation where the soil stays wet for extended periods.