Got a new puppy 3month old

Hey everyone. So I just got myself a new puppy, Shih Tzu Chihuahua mix, from a co-worker. I had dogs before but even then I don't really remember the bby yrs for then.

My new puppy follows me around alot. He's slowly starting to go outside into the grass to do his business, without me having to go into the grass. No accidents inside the house luckily. Hoping house training will be easy.

We're trying to teach him that its ok to be alone for a bit outside. He has a open crate with his toys and bed that he does go in. But once you go inside. He goes to the door and starts to whine and cry alot. 1st day it was a mix of crying and barking.

Its now the 4th day. He's gotten it down to a whine but still stays at the door and hits it sometimes with his legs. The other problem is he's learned to use stuff to climb on top of our washing machines. We just removed the stuff he uses as stepping stone to climb his was up. So now that should help so he doesn't get hurt.

He eats well but if I feed him and walk away to either do something else outside or to maybe go inside, he'll stop mid-bite and run to towards me and only if I happen to pass by his food bowl he'll quickly get another bite and run back to where I am. So read around bout what I can do bout that and basically, I feed him breakfast around 7-8am. If I walk away to do something and he stops eatting to follow me. I do give him a chance to finish his food but if he doesn't. I remove the food until it's lunch time (12pm-1pm) hoping that would teach him to finish his food and not leave it there all day.

Some might say to tire him out before leaving him outside for a bit, but I do. I spent 30mins to an hour running around outside with him, playing with him to the point he's panting, with breaks of course. But once I go inside or anyone in the house goes inside. The whining and hitting the door starts again.

I know new puppies can be hard at first and I'm hopping he'll grow out of it. It's only day 4 with him. Might be a little overwhelm with him right now. So taking a little break for some me time typing this up. Any advice can be helpful.

I use to own 3 Chihuahuas and 1 small terrormix, 1 passed at 13 and the other 3 passed at 16. Its been roughly 3yrs since my doggo passed and decide I was ready to give another dog the same love and care.