Hello,
I can help with a few of your questions, having bought a couple houses, myself.
In lots of places, the typical estimate is to reserve about 3% of the overall sales price for closing costs, assuming they aren't negotiated onto the seller in any capacity. That number, though, can vary based on where you live. As long as you have 25% overall that you can dedicate to the down payment and closing costs you should be fine, but you'd definitely want to verify that with your real estate agent.
Your emergency fund, if it's based off your monthly expenses, should only have to change to cover the change in the expense of the house vs whatever you were already paying. However, you might want to keep some cash in reserve to do some number of repairs or changes once you move in.
Mortgages have fees associated with them. They also have risk--even if you're pre-approved, you might not actually get approved for a loan on any particular property. What that means is you do pay more overall if you take out a loan as the fees will be tacked onto the principal. Also, since there's risk involved in dealing with a buyer who's taking out a mortgage (it might fall through), a seller will often prefer to see a cash offer, to the point they would be likely to take a lower offer if it's a cash offer. Therefore, mortgaging the property will cost you extra in two ways: mortgage fees and higher sales price.
However, the money you put into the property has an opportunity cost. With mortgage rates still in the 3s, you can always invest the money elsewhere and get a much better return. It's up to you and your comfort level.
The best thing you can do while buying a house is talk to a reputable lender and get pre-approved, and then get with a good agent who can answer any and all questions you might have, specific to your locality. Both a lender and an agent together can answer any detailed questions you have. As a buyer, your main concern is finding a property you like in a location you like. Just never forget the old adage when it comes to real estate: location, location, location.