Good news, the Forgotten Password works correctly and changing it in the My Account works correctly ...

Look in the Configuration ... Customer Details ... and check these two settings:
Customer Shop Status - View Shop and Prices
Customer must be approved to shop
0= Not required
1= Must login to browse
2= May browse but no prices unless logged in
3= Showroom Only

It is recommended that Option 2 be used for the purposes of Spiders if you wish customers to login to see prices.
Customer Approval Status - Authorization Pending
Customer must be Authorized to shop
0= Not required
1= Must be Authorized to Browse
2= May browse but no prices unless Authorized
3= Customer May Browse and May see Prices but Must be Authorized to Buy
are they both set to 0?

Look at the Customers in your Admin ... are they all GREEN for Approved or do you see some set to RED?