Short answer: on the whole there doesn't seem to be something wrong in all of this a more explicit answer would require a fatwa. However there might be an exception: I understand your case as the iqamah is done before an imam is appointed (which is incorrect practice). If an imam was appointed and he didn't allow this person performing the iqamah this iqamah is invalid and a prayer follwing it too according some scholars!
Is the prayer valid?
First of all we must know that the only thing that is considered as essential for the validity of prayer in your described situation is whether or not the time for the prayer has startet (yet) or not if this is answered by yes the prayer is valid (technically: this means there's no known reason from fiqh perspective why it should be declared invalid).
Is this right or wrong or even sinful?
Case of praying at home
One basically could start praying right after the first words of the call of adhan if the prayer time has entered.
This means if you were at home once hearing the call of the mu'adhhin you may start praying with or without iqamah. This generally applies for women. But let's say the husband is at home and in hurry he may pray alone or with his wife once he hears the first words of the adhan and the prayer would be valid as said earlier.
However it would be much better to wait and repeat the words of the mu'addhin and start praying afterwards. As this is what our prophet () has recommended.
Fiqh status of the prayer calls
The majority of scholars consider both adhan and iqamah as sunnah. However some scholars even consider the adhan fard kifyah as people should be informed about the beginning of the prayer times. Therefore it is held in mosques. (see for example: Can someone give me examples of fard kifaya?)
Practice of the prophet and his sahabah
As for the case of in a mosque the sunnah is that Bilal (or any of the few other sahabah who were mu'addhins for the prophet()) made both prayer calls: adhan and iqamah so there was always a time gap between both.
He () even recommended two travellers to perform Adhan and iqamah first things first:
"When (both of) you set out, pronounce Adhan and then Iqama and the oldest of you should lead the prayer." (Sahih al-Bukhari)
And it doesn't sound good to do something else in a mosque unless the imam is informed (if there's a regular imam else the community should have a certain consensus on what and how to do, if somebody is doing so against the will of the community it might cause harm and would be sinful).
Basically it is the imam who choses the person that may call for iqamah in absence of a possible regular mu'addhin. As we read in many ahadith that Bilal waited for the prophet () to appear before making iqamah!
Can iqamah be given by somebody else than the person that performed adhan
The simple answer it is generally disliked but not harmful. As this was not the practice at the time of the prophet (). However if the imam allows this it is his right to appoint somebody else!
Discussing the argumentation
As for the argumentation:
"when you fast for Ramadan soon as you hear the adhan for maghrib you eat right? as soon as you hear Allahu akbar in the adhan you can start praying, and hearing and waiting for the adhan to finish is sunnah".
- All this is correct, but nevertheless is not the usual practice at any mosque I know of.
- Even in Ramadan once you hear the adhan you may eat a date or drink a cup of water to break your fast, but you then move to the prayer when about everybody has broken his fast.
- And it is not correct to hinder people who are present during the adhan from the rewards of repeating the words of the mu'adhhin and asking Allah to give our prophet the right of intercession and illustriousness, and resurrect him to the best and the highest place in Paradise (see for example in Sahih al-Bukhari) after the end of the adhan.
It is also not correct to hinder people from performing sunnah prayers (if there are some recommended ones) before the Fard prayer. This sahih hadith for example is considered as an evidence for praying nafl between adhan and iqamah in general.
Furthermore the usual practice is that the imam moves towards the mihrab before iqamah was held or appoints a person to perform it. If somebody acted against this, this prayer can be regarded as invalid due to the sahih hadith:
The Messenger of Allah (ﷺ) said to us: The one who is well grounded in Allah's Book and is distinguished among them in recitation should act as; Imam for the people. and if they are equally versed in reciting it, then the one who has most knowledge regarding Sunnah; if they are equal regarding the Sunnah, then the earliest one to emigrate; If they emigrated at the same time, then the oldest one in age. No man must lead another in prayer in latter's house or where (the latter) has authority, or sit in his place of honour in his house, except that he gives you permission or with his permission. (Sahih Muslim)
and in a shorter version:
"A man is not lead in his Sultanah, and his seat of honor in his house is not sat in without his permission." (Jami' at-Tirmidhi)
- All fatwas I've consulted say that it is the usual practice that the person who calls for adhan calls for iqamah and only the imam has the right to change this.
This fatwa of sheikh ibn 'Othaymeen (video in Arabic) covers the case of a mu'adhhin who calls for iqamah before the appearance of the imam (even based on pre defined rules: for example iqamah is 10 min after adhan) and would start the prayer if the imam didn't come on time. He answers: if he does so without the agreement of the imam or before the imam appears this is haram (illegal) and some scholars said this prayer is invalid. If he did so with the agreement of the imam everything is fine, but nevertheless this person should wait 1 or 2 more minutes for the arrival of the regular imam. The main evidence for that is the hadith quoted above about "giving permission to lead a prayer" by the person who should lead it by default: The host if one was a guest or the regular imam in a mosque.
- As the the mosque has no regular imam this creates confusion and is the main issue in this case.
- As again an imam is recommended to wait before starting the prayer. As he will lead a congregation he should wait an appropriate time for the gathering of the people. This is not given in your case.
All these facts and issues should be discussed in the community especially as they don't go a long with the practice of our prophet (). And they could require a fatwa.
Especially for the case if this individual is imposing his wishes or decisions on the others he seems to be acting sinfully, but Allah knows best.
So for such cases one should declare who should be imam with a certain order (for example based on the hadith quoted above) and this person is the one who gives permission for iqamah.
The "special" case of maghrib prayer
Finally in the case of maghrib prayer the scholars are in consensus that the prayer should be performed soon after the entering of its time (especially in mosques). Some scholars even reject that there's a sunnah before maghrib. Basically the recommended time gap is the time to perform wudu' after the call.
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