Only if it uses a software exploit. Java and Flash are the most common cases, so not having these will vastly reduce the chance of that happening.
Of course there are exploits in browsers and the like but Macs aren't really targeted -- not due to security through obscurity as some maintain, but simply because it's a lot more difficult to pull off. Even if an exploit somehow runs, all system files are locked down with SIP since El Capitan and infections can't elevate privileges as easily as they can in Windows. Certainly no way they can run sudo commands without the users' knowledge; definitely no way they can get close to running a root command.
Saying this is likely to prompt a number of commenters pointing to hypothetical scenarios or the odd previously discovered Safari exploits/root exploits as 'proof' that it's possible; however the point remains that there has not been a single virus in OS X's history and no known virus that used the aforementioned exploits. Lots of malware, sure, but no viruses. That's not to say you should be carefree or think that it's impossible to get infected; of course you should continue to practice safe browsing habits and vigilance. But you can be reassured that you can enjoy your machine without needing to be too paranoid or concerned.

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1) Personally I'd say Safari, though that doesn't mean to say you should stop using your browser of choice.
2) None that I know of. Hypothetically possible with exploits, but then almost everything in the computer world is hypothetically possible.

3) That's pretty much the case!