How to display all fields belonging to current entry?
Field groups are not specific to a particular channel or channel entry. They are just an arbitrary organization within the control panel for ease of use. What you would be looking for are the fields defined for the particular field layout for that entry. I.e.:

{# loop through the fields in the entries fieldLayout Model #}
{% for fieldLayoutField in entry.getFieldLayout().getFields() %}

    {# get the field Model from the fieldId #}
    {% set field = craft.fields.getFieldById(fieldLayoutField.fieldId) %}

    {# print the field handle and the field content  #}
    {{ field.handle }} : {{ entry.getContent()[field.handle] }}

{% endfor %}


By the way, there is an excellent plugin called 'Inspector' written by Adrian McNeal that displays all methods/attributes/properties of an object or variable, which can be very useful for figuring out what your dealing with.

UPDATE: It has been brought to my attention thatfieldLayoutField.fieldId has been discontinued in favor of fieldLayoutField.id. While not tested, if the above does not work, please try the following instead:

 {% set field = craft.fields.getFieldById(fieldLayoutField.id)