It can be done but you will have to go to a portfolio lender and will likely have to pay higher interest rate with more stringent terms. I have the name of a bank that one of my foreign clients used last year. Call me if you want the contact info.
I don't know exactly how Zillows zestimates work but I would imagine it just some sort of algebraic formula that utilizes the assessment, and sales in the area. Zillow can't possibly know what every homeowner has done (or not done) to their individual homes or condos. If you pulled permits for your improvements, they may be reflected in the assessment, and thus may be reflected in he price.
1) Ask seller's agent for the comps they used to support the price. Maybe you are missing some comps.2) If you are concerned about the property appraising, insert a provision into the offer, "Subject to property appraising at or above purchase price," that way you will have peace of mind that the property will appraise.
Evan - I think we may be mixing concepts here. In my area, "Entry Only" means that the listing is posted to the MLS but the listing agent provides very limited service. An "accompanied showing" is a showing of the house to the buyer with the buyer's agent present and the listing agent present as well (thus the listing agent "accompanies" the showing.) Unless there is something very unique about the house that is not readily evident and thus requires the presence of the listing agent, I think that "accompanied showings" are best used for 2nd showings. Buyers find the presence of the listing agent too intrusive. Hope this helps.