In light of the Supreme Court's decision that corporations have first amendment rights it's worth addressing the question - do corporations have rights? Thankfully this is one of the more trivial legal questions (really). The answer is no.
Why do corporations exist?
Imagine you have a couple of extra bucks around and rather than shoving it under your mattress you decide you would like to invest it. You find a nice company in your town who you think is run by nice people and has good prospects and you make them an offer. If they will give you an ownership share in their company (and hence partial ownership of all the profits that are going to roll in) then you would give them the extra money you have laying about.
The company agrees and you don't think much more about it until you find yourself in a court room. It turns out those nice people left town with the company's assets but left behind the company's debts, all of it, much more than you actually invested and you are now personally on the line to pay it all back.
This is how personal property works. If someone, for example, gets hurt in your home you are liable for the damages to them. No one cares if the damages are more than the house is worth, the two things are unrelated. You own the property so you are responsible for what happens with it. In the same way when you are owner of a company you are responsible for what the company does, including any debts it incurs, even if you share that ownership with others. (O.k., I'm over simplifying, see here for some more details).
And for most of human history this is more or less where things stood. But over time it was recognized that there was a problem with this ownership approach. It made people unwilling to invest their money because they could be held responsible for the debts of the company they invested in. The property laws were slowing down the growth of the economy.
This led to the introduction of limited liability (amongst a number of other legal innovations). Governments passed acts giving companies, primarily (but not exclusively) corporations, the right to form and get investors and for those investors to not have any responsibility for the acts of the corporation, this included everything from debt to negligence claims. So if the company went bankrupt with large debts (or, say, killed tens of thousands of people), the worst the investors suffered was the loss of their investment.
Limited liability is a privilege, not a right
But notice, there is no inherent right to limited liability. In fact, quite the opposite. The government is offering a deal. In return for helping the economy the government offers a pre-approved liability waiver. But the terms of the deal are for the government to decide. If the government wants to levy heavy taxes on corporations, limit their activities, set expiration dates on the corporation's existence, etc. that is all within their right.
So no. Corporations are not people. They do not have rights. They are a legal fiction created for the benefit of the citizenry and as such governments are free to set any and every restriction they like on the formation of corporations. If citizens don't like it they can change the law or they can form partnerships or sole proprietorships and accept full legal liability. But if they want the benefits a corporation brings then they need to accept whatever limits governments sees fit to apply. Limiting political donations, which started us off on this journey, is but the most minor of those limitations.
Where to learn more
The story of corporations is a truly fascinating one. John Micklethwait and Adrian Wooldridge wrote a short but very fun book called "The Company: A Short History of a Revolutionary Idea" that I highly recommend.
If you want to learn more about corporations in American legal history then I recommend "A History of American Law: Third Edition" by Lawrence M. Friedman. It's well written and although a survey of all law in America (and many of the key legal personalities) I think it fair to say the author's thesis is that more than anything else business drove the development of law in America so most of the book focuses on how the law and business interacted. Note however that having some background in English common law would make this book an easier read but it's not strictly necessary.