First just let me dissuade you of this "studying programming" idea. For one thing, none of the universities actually do a very good job of it; and for another, there's no degrees issued in "programmer" that I am aware of. Now, you can take college/university level programming courses, but you won't find a lot of those and you will likely have difficulty converting that training into a worth-while career. There are too many highly qualified "programmers" in China and India willing to work for $4K a year or less. If your interest is merely to learn how to write code as a hobby, then please do sign-up, it can be fun and doesn't require much in the way of prerequisites.
If what you are interested in is more along the lines of computer/computational science (CS) or engineering (CE), then here's my responses to your four basic questions:
1. Take one of the college entrance exams and see where they rate you in the math department. Generally low scores in math is not too big a deal provided you are capable of thinking logically. Lots of folks who do poorly in high school calculus and algebra actually do well in formal logic (rarely taught in k-12). If you can get through logic and algebra 101 at the college level, you have a fighting chance in CS/CE and a follow on career. You don't have to be a math wiz in most CS/CE specialties.
2. Take at least two more classes in statistics! Particularly Bayesian statistics
(actually quite simple). Statistics and logic are just good knowledge for any citizen of the world to know and they are used extensively in CS/CE.
3. Yes. If you prefer not be challenged, don't bother with CS/CE. Not because it's math intensive, but because it is challenging in its own right regardless of what you think of the supporting course work. CS/CE careers are no piece of cake either. If you don't like to be challenged, consider doing something less challenging than any of the sciences or engineering disciplines.
4. No. If you don't have at least a high school level understanding of algebra, you will probably fail early in CS/CE. You don't have to be a genius at it. If you passed a couple of algebra courses in high school you shouldn't have much trouble passing a college level algebra class.
Now after saying all of that, I have to admit that I am a high school drop-out who has been working as a software engineer for 23 years. I have taken a lot of college level courses over the years and I read ACM, IEEE and other technical material every day. I also work with masters and phd level folks on a daily basis. My math skills suck by comparison to many of the folks I work with, so it's a daily challenge for me and as a result, I probably have higher than average skills in math today (compared to what would have been my graduating class anyway).
The cool thing is that math gets really interesting when you can apply it to real-world problems that affect your career performance. Trust me, if you have what it takes to be a good computing engineer or scientist, you'll eventually come to appreciate the beauty of formal logic and Bayes' theorem.
BTW: I worked in construction for 20 years before converting my electronics hobby into a software engineering career. All of my college level course work has been as a "mature student". I firmly believe in life-time learning. If you are not currently a student of something, you should be. It will help extend and improve your life in many ways.