Question: If submit a mortgage pre-qualification application, does my credit score go down? Does it trigger a hard credit check? Or do the hard credit checks only come when the mortgage process goes further than pre-qualification?

Answer: 
Your question is loaded with some very specific terminology about certain steps in the mortgage process. It may help to zoom out and describe the full mortgage process as it is typically carried out in the US.

Many lenders offer a **pre-qualification** process which involves no commitment by either party and is essentially meant to be an estimate. A pre-qualification is quick and basically involves looking up a generic rate and loan size based on (customer-provided) basic information about debt, income, and credit history. Essentially, it's the lender using your information to look up a loan you *might* qualify for, in a very generic schedule - you give them these basic data points, and they tell you "you are pre-qualified for up to $400k at 4.9%" or whatever the relevant numbers are.

True pre-qualification typically **does not** involve an actual credit report being pulled, it's done on customer-supplied data. Lenders are OK with this because of the lack of commitment involved in the process. Effectively, for a lender, it's more of a sales tool than anything, since the process gets the personal data of an active shopper in the hands of a mortgage sales person at the lender. A lender is more than happy to spend a few minutes (or have a computer system spend a few milliseconds) churning some numbers to get that prize.

Different from pre-qualification is **pre-approval.** Pre-approval for a mortgage *does* involve a credit report being pulled, and it will be a "hard pull" which will show up on your credit report. It also involves checking other data sources - a lender will want to have some proof of income beyond just you telling them a number, and so on. Lenders will want to pre-approve customers before making any sort of commitment (i.e. a commitment letter for a specific purchase). The outcome of the pre-approval process is usually more specific than the outcome of a pre-qualification, i.e. "you are approved to buy a house for $300k with a $50k down payment and an interest rate of 4.9%, assuming no changes in income or credit score etc."

Upon finding a specific home you wish to purchase, the lender will process your actual loan application through underwriting, which will generate a **commitment letter** - an indication that the lender will provide you a loan for a specific amount for a specific house with a list of conditions or other details. A commitment letter is essentially a way to turn your pre-approval into a binding deal for a specific house.

As long as there are no significant changes that invalidate the pre-approval, and the conditions of the commitment letter are met, the bank will proceed with the loan as outlined. The lender will usually pull your credit report again immediately before closing (usually within 48 hours) to validate that you haven't done anything foolish like stop paying your other loans, bought three new cars, or otherwise changes your finances significantly. This will be another hard pull, and depending on how much time has passed since your pre-approval, it may or may not hit your credit score a second time. The lender may also want to see additional proof of income (pay stubs) and/or up to date proof that you have enough cash on hand to close the loan (bank account statements showing sufficient balance). Pre-approvals and commitment letters both typically have expiration dates, and if the process stretches beyond those dates the lender may want to repeat some or all of the steps involved.

It's important to note that some lenders don't actually do pre-qualifications, so the first step with them will be a pre-approval, with a credit pull. When in doubt, as the lender you intend to use what their process is. They will typically be more than happy to walk you through all the steps involved and give you a chance to ask questions.
You will likely get a hard credit check. You can ask the bank if they can give you an estimate of the mortgage amount and interest if you give them a credit score and income. But some won't do that. 

In most cases the bank will run the credit check and give you a pre-qualification based on the results. Anything in-official (without running your credit) won't give you a pre-qualification. At best you'll get a verbal agreement that if your credit score is what you say it is when they run it, they will give you a pre-qualification in the amount they told you.