I'm going to answer your questions out of order.
Emergency fund: Depending on how conservative you are and how much insurance you have, you may want anywhere from 3-12 months of your expenses on hand. I like to keep 6 months worth liquid in a "high-yield" savings account. For your current expenses that would be $24k, but when this transaction completes, you will have a mortgage payment (which usually includes home-owners insurance and property taxes in addition to your other expenses) so a conservative guess might be an additional $3k/month, or a total of $42k for six months of expenses. So $40-$100k for an emergency fund depending on how conservative you are personally.
Down payment: You should pay no less than 20% down ($150k) on a loan that size, particularly since you can afford it. My own philosophy is to pay as much as I can and pay the loan off as soon as possible, but there are valid reasons not to do that. If you can get a higher rate of return from that money invested elsewhere you may wish to keep a mortgage longer and invest the other money elsewhere.
Mortgage term: A 15-year loan will generally get you the best interest rate available. If you paid $400k down, financing $350k at a 3.5% rate, your payment would be about $2500 on a 15-year loan. That doesn't include property taxes and home-owners insurance, but without knowing precisely where you live, I have no idea whether those would keep you inside the $3000 of additional monthly home expenses I mentioned above when discussing the emergency fund.
That's how I would divide it up. I'd also pay more than the $2500 toward the mortgage if I could afford to, though I've always made that decision on a monthly basis when drawing up the budget for the next month.