A prayer begins with takbir (saying "Allahu akbar") this takbir is called: takbirat al ihram or takbir for the beginning of prayer. At this moment you start your prayer and leave anything else therefore it's called by scholars in Arabic takbirat al ihram as you leave anything which has to do with donya and face Allah (this is just one of the reasons).
When we say fard prayers are sobh/fajr two raka'a dohr, 'asr and 'isha are 4 raka'a and al-Maghrib 3 raka'a: this means the prayer which has been started with the takbira will have to be ended this means you need to leave the prayer and therefore you have to do salam at the end of the last raka'a according to each prayer: you do this by slightly turning your head right and then left (according to the majority of the sunni-madhabs) and saying at least: as-salamu 'alaikum while turning your head! but most say or agree that once you turned your head slightly to the right and said as-salamu 'aleikum you left/finished your prayer!
As a fard (obligatory) prayer is something you have to do and a nafl or sunna (voluntary/optional) prayer something you are recommended to do they have different legal status therefore you have to consider them as independent prayers and therefore you can only pray a sunna prayer after doing salam for the fard prayer. And for a sunna prayer you don't need to do neither athan nor iqamah.
And if you passed by a hadith saying that prayers have been combined: This means only that two closed prayers have been prayed at the time of one of them (see here).
About the intention: some scholars say it's a bida'h to pronounce your intention as the place for it is in the heart. This means if you want to pray 'isha' you think of it and you do it. But the niyya/intention has to be made right before takbirat al-ihram (the takbir at the beginning of a prayer): Niyya is an intention followed by the action (or better an making an intention beside a co-instantaneous action) which you intended to do.
Some people still complicate the niyya matter by saying "i intend to pray such a prayer in congregation with (such) the imam and it will be of such raka'a etc." so what if you by mistake said i'll pray dohr while what you really are going to pray is 'asr? Is your prayer valid as you spoke out a false niyya while you really intended to pray 'asr?