When and in which country is it permissible for me to pray shortened (qasr) and combined (jamʿ) prayers, taking into consideration my travel between Egypt and the Netherlands, and must I adhere to this throughout my short stay in Egypt, especially if it includes Ramadan and Eid?

If you intend to reside in a country for four days or more, you are obligated to pray in full and perform the prayer at its prescribed time. You are not permitted to avail yourself of the concessions of travel. Shortening the prayer during travel is a Sunnah (definition), not an obligation, and combining prayers is a concession during travel.