Should I stash my house down payment in the same place as my emergency fund?

No, keep them separate, both to protect each purpose and because they have different timelines. Mixing them makes it easy to convince yourself the down-payment money is really emergency money, or vice versa, and you can end up short on one when you need it. Put the down payment in its own labeled account sized to your target and time horizon; if you are buying within a couple of years, safety wins, so use a high-yield savings account, money market fund, short CDs, or Treasuries rather than stocks that could drop right before closing. Meanwhile your emergency fund stays fully intact for true surprises. Separate accounts also let you track progress on the home goal without it distorting how much safety net you actually have.