"What rights is Trump actually proposing to remove from women?"
Very basically, the right to bodily autonomy.
It's not just about abortion, but, riding on the pro-life wagon he's intending to defund Planned Parenthood. Only a tiny fraction (something like 3%) of PP's services are abortion related - and none of them publicly funded. What Planned Parenthood does do is offer, among other things, birth control and birth control education, STI testing and treatment, and cancer screening, to both men and women. They do this regardless of a person's insurance status or ability to pay. Trump and his administration also intend to repeal the Affordable Care Act, leaving millions with no health insurance and no alternative.
People will die because of this, many of them will be women.
Madonna is crass and tactless, and I don't personally care for her, but if you care to look past the words she uses to find out what she's actually trying to say she's very often right. (And she didn't tell anybody they should blow up the White House, ffs.)
Based on the well-known people I follow on Twitter, there were a lot of 'celebrities' at marches around the country, and none of them were there to give speeches or court publicity. They were there because they're all human people who are, and/or care about, women.
I'm in the US, and I can't speculate on why there were marches in other countries - the people involved will have their own reasons and I don't expect them all to be the same.
The fact is the Trump was 'democratically elected' by only the narrowest of definitions - and that definition does not include the popular vote. The President is meant to answer to the American people, not the other way around: He (along with every other politician) is an employee, nothing more and nothing less. I feel like the marches here were intended to bring attention to those two related points.