So is it true that women need to have a mahram with them whenever they go somewhere and how strictly is that enforced?
There's this really big misconception that women here are not allowed to leave their houses or go anywhere without a mahrem. It's not true. Saudi really caters to the women here. Where they may lack giving them something, they replace it with something else (eg. they can't drive, but allowed to sponsor foreign drivers). So when women are out, say they want to go to the mall, the malls here allow women and families (husband + wife + kids) to shop freely, but if a group of teenage boys or any man without a family wants to shop as well, they are denied entry. This is to protect the women from having to deal with harassment from single men. Most large establishments where people like to go to waste time and loiter around are like that. However, places like grocery stores, pharmacies (where you're actually going cuz you're in need of something) are not. Secondly, having drivers or using taxis are utilized by many women here to go to work, school, and other necessary places. They don't need to have a mahrem with them to utilize them. That goes without saying though, if you want to go out and your dad says no, then you can't go out

I think that runs in any household.
So then, what is this huge rumour all about then?? THe rumor is about women who want to leave the country. Any woman who wants to leave, has to have permission from her mahrem. What that means is, there is actually a governmental online system, where the mahrem fills out an application that basically says "My wife/daughter is allowed to leave the country with my knowledge" So whenever they are flying or crossing the border, the mahrem is notified that they have left via a text message (if he is not with them). Also, there are hotels here that require one to show his "family card" to prove that the female with him is his mother/sister. It's a governmental issued card that lists the names and relationships of the family members in the first degree. This is to limit any hanky panky that may go on if a woman and a man are trying to get a hotel room and commit some haram lol. Every individual household has their own rules though, so some families may be a lot more strict in allowing female members to leave or go anywhere without a mahrem. However, it's purely individual and it's not really enforced by the government. I also want to add that there are many nonMuslim women who reside here on their own. NonMuslim westerner women who are sponsored by companies or private international schools are given housing, drivers, and anything they need really to live here on their own. When they want to leave the country, then it's their sponsor who has to give the permission to do so in that situation. They can't go anywhere without it.
Do they allow me to move there and become a citizen I wanted to live there but my dad says they don't let people immigrate also if I live there I can go to Makkah at any time for ummrah right?
You can't become a citizen that easily, but you can move here with reason (ie. for work/business purposes) or if you're the son/daughter of someone who is sponsored to come there. Even women/men who have been married to Saudis for years have difficulties getting the citizenship. However, Saudi is in the works of making it easier for people to come and stay (with a price), they're thinking of having something similar to the US green card. Also, rumor has it that they may even be opening Saudi for tourism as well. My friend's father was given the opportunity to get the Saudi citizenship. He's worked there for years and years and I guess he was highly respected so they offered it to him, but he refused it. So it can happen, but you really, really have to earn it, and most likely it will be after years and years of residing there.
To go to Umrah, you have to get a visa to travel there first. I think they are more lenient with allowing people to come for Umrah whenever they want. (Tip: Come during the winter months

Dec-Feb). For hajj though, it can be more difficult and a lot more expensive because you have masses and masses of people coming from every country. There's actually a limit on how many people are allowed to come from each country (including Saudis I think) so that they can ensure every country is given a chance to perform hajj.