You're income certainly isn't "normal" and some banks might shy away but this is a personal business and if you have a track record, I think you can find a bank to say yes.

Can I give a tenant written 30 day notice on June 1 (in person or taped to the door) for move-out on June 30? 

I used my personal cash for both , but I’m looking to purchase more with lending.

I personally own properties in the counties of Orange, Los Angeles, Riverside and Kern.

However, the money I’d save by not paying $3500/month mortgage is $42,000 after 12 months – thus recovering the money paid on the capital gains tax.QUESTION: would you advise selling the rental to pay off the new home and take the $1400/month rental income loss (and gain $3500/month savings) or would you advise keeping the rental?

If you could go in person that helps. 

Becoming a landlord is more work than the average person thinks.

LLCs offer no tax benefits, they mitigate liability and the fact that this is on your personal property and you'd be renting to yourself, there's no liability to mitigate.

It is our personal house and we're in the process of moving back to the east coast (New England area) to a higher COL area.

It has not been well maintained over the last 19 years with a string of just bad renters and the person I had managing it just didn't do good quality repairs or work to it.