Does the registrant still have common law rights over the mark?
Maybe. According to Yospin Law,
Just because a mark is listed as “dead” at the USPTO, does not mean it is available to use without registration, or to register. It’s possible that the registrant of the trademark abandoned the federal trademark registration, but is still using the mark – and so still has common law trademark rights.
Does this mean the registrant can never register this mark again?
No. It does not mean the trademark can never be registered again. However, a new application for registration of the mark must be made. This registration is completely separate from the previous one - it's effectively registering a new mark that happens to be identical to an old one, without the problems that would normally cause.
Can another entity register this mark for the same purpose?
Possibly. If it remains in use under common law rights, you would face an uphill battle to successfully register it for yourself. If it is truly abandoned, you can apply for registration, but as Beth Hutchens discussed, this is not a simple matter of filling out the form and celebrating your new acquisition. Here is a handy brief summary, but for any actual case, your first step should be to consult a patent and trademark attorney, particularly one experienced in the jurisdiction you intend to use the mark in.