Is it true that foreclosed homes usually sell for a discount? I mean, the bank is in a hurry to recover whatever money it can, so it makes sense they would be willing to take a hit on the sale price to sell it off quickly. If yes, is there a website for foreclosed properties that are offered for sale?
"Usually"... I think that's overstating the case.
You CAN get a bargain (especially if the place is in not-so-great condition), but not every foreclosure will be a good deal even if it is priced well below its most recently appraised value. As the buyer it's your responsibility to determine whether it's priced well or not, and to decide whether you're willing and able to repair its deficiencies after you buy it. The same's true when purchasing any house; foreclosures just make it more likely that there are problems and (hopefully) wind up being priced to allow for them.
I don't know of a single website which lists all foreclosures. Some of the home listing websites do have a "show me foreclosure listings" filter, and I'm sure that the better tools available to real estate agents can select these. But if that's the direction you're interested in going, you should be looking at distressed properties generally, NOT just foreclosures; you may get a better deal, in the long run, by going for the one that has been mechanically maintained but is just plain ugly rather than the one with a pretty skin whose heating system hasn't been serviced for the last decade.
Do your homework, shop around, don't fall in love with any one house... all the same rules apply at this end of the spectrum just as strongly as they do in the mid or upper ranges. Perhaps more so.
No, it is not true.
It depends on the market, the banks' inventory, the original debt that was owed, etc etc. The banks generally want to recover their money, so in case of underwater properties they may end up hold a property for years until prices bounce back (as it happened during the last crisis when many houses were boarded for months/years until banks put them back on the market hoping to sell at a price that would allow them to recover their losses).
That may depend largely in which country you are in, the legislation in that country and the state of the economy and property market (more specifically) at the time of the foreclosure.
In Australia, where we do not have non-recourse loans (except in SMSFs) the banks are obliged to recoup as much as possible for the mortgagee, however they would not hold on to the property indefinitely, as that could cause other problems and they have to return the mortgagee portion of the funds back to them (if there is any funds left after the bank takes their portion).
In 2008, when the property market here was weak we had bought some foreclosure houses and were able to get them 20% to 25% below what they were selling at the year before. If there was a forclosure in today's strong market in Australia (and especially in Sydney), I dought you would get much of a discount at all. So it may largly depend on the demand and supply at the time of the forclosure.