I am not Muslim and have entered at least 30 Mosques in my travels. From India and Malaysia to Egypt and Nairobi I have entered many. You have to take your shoes off and be respectful as it’s a place of worship. You have to respect like you would be in a church. As far as women go, dressing with modesty is necessary, a friend I was with was asked to cover her hair with a hijab out of respect.
Not much different than churches in the days of old where women used to enter with their heads covered, as you can still see in some Mexican villages which are very Catholic. If a woman is dressed with a short skirt, shorts or a tank top or with a low cleavage, well she might run into trouble getting in. If she dresses correctly for the occasion she will get in without a problem from my experience. As long as you show respect I see no problem in getting in.
Many of the Mosques I have been to are extremely beautiful, they vary also, the ones I entered in Kuala Lumpur, Malaysia were very modern looking buildings especially on the outside others were very traditional blended with their own unique culture. They are all beautiful and I was treated kindly and respectfully in all.
These are some of the Mosques I have entered:
Jamia Mosque in Nairobi
Muhammad Ali Mosque in Cairo
Blue Mosque in Cairo
Kampung Laut Mosque in Malaysia
Kuala Lumpur Malaysia National
New Delhi India: The Jama Mosque
The Masjid Sultan Mosque, downtown Singapore
Malacca straits mosque, Malacca