I think Jaide and I are having a similar issue in our new house. We moved here middle of last year and there's been some fairly weird stuff. Our bedroom closet door opens on its own. It has a firm latch, can't vibrate open, and it's where we put the dirty clothes hamper. Right up next to the door. Last week I went to grab something and the door was 6-7" open. I have literally tried everything to pop it open (stomping, even straight out pulling on it), and it just does not open.
The other odd thing, if I open the closet door to get something, the cats hear it and immediately run in there to be as big of a pain in the ass as possible to get them out. You'd think I was hiding a kilo of sweet columbian catnip in there. When the closet door opens on its own, they never go in. They stay away from it.
Last night I was so exhausted I slept through the night (which is abnormal for me, I usually don't get a full nights sleep), but Jaide said there were a lot of weird noises. We do have cats, but they were both upstairs and "on guard". We've had this happen a few times. Sometimes occasionally makes them go batshit (... catshit?? haha). Jaide will be cuddling one of them and they'll suddenly become alert and stare intently at the top of the stairs to our bedroom or in a corner in the downstairs living room. We've had these 2 cats for 10+ years in multiple houses, and we've been here since July so they are acclimated. This is weird behavior for them.
As for communicating, go for it. Use whatever method you feel most comfortable with. If there's an actual entity, and they want to communicate, they'll make something known. Do your best not to be afraid or show fear. A lot of damage attributed to entities are actually self harm.
Be firm, like Yazata said, if you want to banish whatever is in your house. YOUR MINDSET MATTERS (might even be the most important thing). Even walking through with burning sage could be beneficial. Some of these rituals are less about magic and more about guiding your mindset to a specific place, calm you down, give you the mental energy and focus to deal with the problem.
We've used tarot cards and a spirit box that I built myself to try and communicate at this house. I won't rehash the technical details (there's a forum post on it), but suffice to say I live in an area that at the best of times can pick up maybe 2-3 FM radio stations, so most of the frequency being swept by the sprit box is 100% static. I also randomly hop frequencies, so I don't hit one of those 2-3 FM stations for more than 1 hop in a row.
This should result in absolutely nothing coming through the spirit box except mostly static and the occasional garbled half of a word. Instead, we get full phrases and sometimes sentences out of it. Since I built it and the software that runs it, I really can't explain it. There is nothing on the device, in the software or in our geographical area that would explain why it actually works.
I think we've managed to communicate through both methods. I still don't know who or what is here. In fact, I'm almost positive there are multiple entities here. From some of the responses, some at least try to appear as if they were previously living, but there is one "voice" that doesn't appear human. It has literally said "Paimon" twice. I have, actually, been working with Paimon, but I'm not convinced this entity is telling the truth.
Saying "Paimon" really brings up whether or not these are essentially egregores. The name was clear, not garbled or staticy. Maybe the work I've been doing with King Paimon is self manifesting and I've created that entity with my own energy. I don't know. Post manifestation, these entities seem to have a mind of their own, so I'm not sure how they are created is really all that important.
We have not done any banishing yet, but I did take a break from trying to communicate. I think in my mind I didn't want to seem ... controllable. Communication will be on my terms. This is our house now, and I'm not going to give over control to some unknown. Maybe I'm a silly little mortal and I'll end up regretting that, who knows. I want to try to find out more, so ultimately I am inviting whatever it is to stay for now.