I know the possibility of the Imam giving Adha, but can he give the Iqama as well as lead the prayer? I asked him for the reason, he said, what if the people behind can't even say Iqama? Then he can do so.
"It's permitted to give Adhan and Iqama, and leading the prayer..."
Yes, it is permitted without dislike or impropriety for the one who gives the adhan and/or iqama to also lead prayer. (Ibn Abidin, Radd al-Muhtar; Tahtawi/Shurunbulali, Hashiyat Maraqi al-Falah)
Anybody can perform adhan, iqamah and lead a prayer so there' s no exclusion. The only point is that an imam would gain more rewards than anybody else if he lead the congregation in the best manner therefore it would be recommended for him to follow the sunnah of our prophet end leave other's perform adhan and iqamah to gain extra rewards.
This fatwa clearly say's that an imam can perform iqamah. Note that this fatwa in Arabic says that the person who perfroms adhan and iqamah should be the same as this was the sunnah in the time of our prophet, but an exception could be if the person got a persmission from the imam to perform the iqamah instead of the "regular" muaddhin.
This fatwa (I think the same as the one quoted by IBRA) is saying that an Imam can perfrom iqamah and adhan also and includes the quote from mawahib al jalil and one from Imam an-Nawawi's al-Majmu' saying that there is consensus among Muslim scholars that this is allowed.
أجمع المسلمون على جواز كون المؤذن إماماً واستحبابه، قال صاحب الحاوي: في كل واحد من الأذان والإمامة فضل
The Muslims are in consesnsus that it is allowed for the muaddhin to be an imam and recommeding it. The author of al-Hawi said: in each of adhan and leading a prayer there's a reward.
Note that when quoting adhan often also iqamah is meant as usually both are perfromed by the same person!
This fatwa links to an other fatwa explaining why neither our Prophet nor the rahidun caliphes perfromed adhan and iqamah ... the major reason was that they had more important tasks. Quoting that at-Thirmidhi compliled a hadith saying that our Messenger performed adhan during a journey, but ibn Hajjar al-'Asqalni qualified this ahdith as da'if. And the same fatwa quotes a saying of Omar:
قَالَ عُمَرُ: لَوْلَا الْخِلَافَةُ لأذنت
Omar said: if there was not the (task of) caliphate i would have perfromed adhan (myself).