I am, Insha Allah, going to Umrah this summer and I would like to clarify few points. If I want to do several times (once in the morning, once in the after noon, once in the evening) the Tawaf and the ritual of Safa wa al-Marwa while I am there, would that be a problem? If so, can I have a shower in between or not? I will be staying in Makkah for 6 days so I want to do the rituals as many times as possible.

Praise be to Allah, the Lord of the World; and may His blessings and peace be upon our Prophet Muhammad and upon all his Family and Companions.

Performing Umrah more than once in a day is permissible, but it was not the practice of good ancestors. They used to make a separate journey for every Umrah. Thus, being busy in remembrance of Allah, performing prayers, and Tawaf is better as the Prophet (Sallallahu Alaihi wa Sallam) and his companions did than making a new Umrah.
However, if one performs more than one Umrah, his Umrah is correct but he has to enter in the state of Ihram from a place that is out of Haram like 'Taneem'.
Taking bath for Ihram is permissible not compulsory. Ibn al-Munzir said: 'All the Muslim scholars have consensus that Ihram without taking bath is permissible and it is not an obligation. Know that taking a bath is likeable due to the practice of the Prophet (Sallallahu Alaihi wa Sallam) and his companions.
Allah knows best.