Salamun Alaikum Chadiga,
The purpose of any calendar is to understand and to mark time. The Quran recognises different calendar systems.
For example in Surah 18:25, we note some people that said:
"And they remained in their Cave (for) three hundred years, and (some) add nine"
The difference of 9 years noted between 300 and 309 years, is the difference between a solar and a lunar calendar.
However, the Quran informs believers what the new moons are for and they are not just for pilgrimage.
Verse 2:189 says they are:
(a) To mark the months of pilgrimage and
(b) To mark periods for mankind (mawaqitu-linasi)
If you note, 2:189 follows the verses about fasting. Therefore by implication, even fasting would be determined by new moons.
Hence, new moons have been suggested as indicators for mankind not only for determining key events (like Ramadan and Hajj) but also for general time calculation as well.
According to the Quran, the 12 month system (based on lunar months) is recognised, four of which are sacred (9:36). This is referred to as the correct 'deen' (system).
I feel that whenever we sincerely attempt to understand Quranic narratives, that we keep its context and immediate audience in view. This helps give us a better perspective.
For example, the Arabs of the desert had no recourse to digital watches or modern technology. Therefore, the Quran spoke to its primary audience with what was available to them. Much in the same way, the Quran was talking to the Arabs with the months they recognised. It was not teaching them how to name their months but informed them of key events to note given the name of the months they knew.
'Ramd' does mean hot, scorching heat or intense heat felt on the ground, but many months in the year are hot in the middle of the desert. I personally wouldn't look too much into the literal meanings of the month other than these were months that the Arabs recognised and the Quran merely made use of them to make the message clear.
Furthermore, I am a little bit confused with the translation you are using. For example, you say:
"The sun and the moon are used for calculation (the time)." (Quran 55.5)
This is not what the verse says. The verse says:
"shamsu wal-qamru bi'husbaan"
'Generally' when a preposition 'bi' is attached to a noun in this way as in 'bi'husbaan', it indicates 'by' or 'using'. It implies that something A is using something B.
As a crude example: To say bi-taayara means 'using' or 'by' a plane (taayara) as in 'I'm travelling 'by' plane' or 'I'm travelling 'using' a plane'
Therefore, the verse seems to be saying that the sun and the moon are 'using' a calculation which indicates to me that the sun and moon follow a certain course, or some sort of fixed calculation.
This does not mean that the sun and moon are 'used' for calculation.
So yes, I personally feel the Quran is marking key events and indicating use of a lunar calendar system of 12 months to identify these events. (correct deen) However, I don't think this amounts to a prohibition of any other calendar system. This is as long as key Islamic events are noted and used in the months they occurred based on the 12 month lunar calendar.
I suppose this is no different than the system that we make use of today.
I hope this helps, God willing.
Joseph