Is there a web service exposed to get all cities in a country

No. OpenStreetMap has the raw data from which you could extract such a list yourself, but there's no ready-made web service that does that for you.

It is possible to go a little way in your direction with the Overpass API; this allows you to ask for "all objects tagged place=city within the boundary described by the OpenStreetMap relation with the ID x". This requires that you first find out which relation you are interested in.

Say you wanted to find all cities in Moldova. Google can be used to find out that the relation id you are looking for is 58974. Add 3600000000 to that number and use the following Overpass query:

<query type="node">
  <has-kv k="place" v="city"/>
  <area-query ref="3600058974"/>
</query>
<print mode="body"/>

This returns the OSM XML for all city nodes in Moldova.

However, requesting such information live from the database every time you need it is a terrible waste of time; you really want to compute such a list ahead of time and then use that.