Since the "no trading/selling" rule is your own house rule, you're going to have to determine this yourself.

My own opinion is that you should probably allow the auctioning off of mortgaged property.

Why? Because the only thing the standard rules say you can't sell is improved properties (property with houses/hotels). It explicitly allow the selling/trading of mortgaged property, after which the "interest" on the mortgage comes due.

From the offical Hasbro rules book (pdf):

SELLING PROPERTY… Unimproved properties, railroads and utilities (but not buildings) may be sold to any player as a private transaction for any amount the owner can get; however, no property can be sold to another player if buildings are standing on any properties of that color-group.

and later under "Mortgages":

The player who mortgages property retains possession of it and no other player may secure it by lifting the mortgage from the Bank. However, the owner may sell this mortgaged property to another player at any agreed price. If you are the new owner, you may lift the mortgage at once if you wish by paying off the mortgage plus 10% interest to the Bank. If the mortgage is not lifted at once, you must pay the Bank 10% interest when you buy the property and if you lift the mortgage later you must pay the Bank an additional 10% interest as well as the amount of the mortgage.

So since your "bank auction" is your house-rule substitute for selling directly, it only makes sense to maintain the same conditions, and allow auctioning of mortgaged property.

Since the "no trading/selling" rule is your own house rule, you're going to have to determine this yourself.

My own opinion is that you should probably allow the auctioning off of mortgaged property.

Why? Because the only thing the standard rules say you can't sell is improved properties (property with houses/hotels). It explicitly allows the selling/trading of mortgaged property, after which the "interest" on the mortgage comes due.

From the offical Hasbro rules book (pdf):

SELLING PROPERTY… Unimproved properties, railroads and utilities (but not buildings) may be sold to any player as a private transaction for any amount the owner can get; however, no property can be sold to another player if buildings are standing on any properties of that color-group.

and later under "Mortgages":

The player who mortgages property retains possession of it and no other player may secure it by lifting the mortgage from the Bank. However, the owner may sell this mortgaged property to another player at any agreed price. If you are the new owner, you may lift the mortgage at once if you wish by paying off the mortgage plus 10% interest to the Bank. If the mortgage is not lifted at once, you must pay the Bank 10% interest when you buy the property and if you lift the mortgage later you must pay the Bank an additional 10% interest as well as the amount of the mortgage.

So since your "bank auction" is your house-rule substitute for selling directly, it only makes sense to maintain the same conditions, and allow auctioning of mortgaged property.

Since the "no trading/selling" rule is your own house rule, you're going to have to determine this yourself.

My own opinion is that you should probably allow the auctioning off of mortgaged property. Why? Because the standard rules don't prevent selling/trading of mortgaged property, they simply say that after mortgaged property changes hands, the "interest" comes due. And since your "bank auction" is your house-rule substitute for selling directly, it only makes sense to allow auctioning of mortgaged property.

Since the "no trading/selling" rule is your own house rule, you're going to have to determine this yourself.

My own opinion is that you should probably allow the auctioning off of mortgaged property. 

Why? Because the only thing the standard rules say you can't sell is improved properties (property with houses/hotels). It explicitly allow the selling/trading of mortgaged property, after which the "interest" on the mortgage comes due.

From the offical Hasbro rules book (pdf):

SELLING PROPERTY… Unimproved properties, railroads and utilities (but not buildings) may be sold to any player as a private transaction for any amount the owner can get; however, no property can be sold to another player if buildings are standing on any properties of that color-group.

and later under "Mortgages":

The player who mortgages property retains possession of it and no other player may secure it by lifting the mortgage from the Bank. However, the owner may sell this mortgaged property to another player at any agreed price. If you are the new owner, you may lift the mortgage at once if you wish by paying off the mortgage plus 10% interest to the Bank. If the mortgage is not lifted at once, you must pay the Bank 10% interest when you buy the property and if you lift the mortgage later you must pay the Bank an additional 10% interest as well as the amount of the mortgage.

So since your "bank auction" is your house-rule substitute for selling directly, it only makes sense to maintain the same conditions, and allow auctioning of mortgaged property.