To refince my house, do I need any type of down payment? And if yes, do you know how much is required.?

It usually is not stated as a down payment in a refi. They determine the maximum amount they will lend based on several factors including your current financial status/income; your credit report; an inspection and appraisal of the subject property. You can take less if you want so don't think what they quote is what you must take. However the reverse is not true, that is usually the max they will loan.

They will charge fees for the appraisal & inspection unless you have one dated within the past 90 days or so. They will charge a fee to do your credit and there will be fees for reconveying the old loan and writing a new loan. There may be loan processing fee.

The reason people refi is usually to take advantage of lower rates. The lender knows they are giving up a more valuable asset [a performing loan to them is an asset] for a less valuable asset [one paying them less income] Who would want to do that? So there are fees. Going with the same lender may or may not be to your advantage. Go to a site like bankrate.com, don't give them your personal info but you must provide zip code/location and proceed to the loan rates. Find the ones for refi. Drop down or expand the section for fees. Print it out and compare it to what your lender says they will do.

If you are refinancing with the same mortgage company,there may not be any fees. Another company

may have certain charges. There could be mortgage insurance premiums to pay.

No, you can refi to 80% of the appraisal, your "down" is the 20% equity.