It is honestly really depressing and sad that so many people, including the OP and several anwers' authors themselves, are so eager to label awkward usage of Internet communication as "creepy". It could get unusual, or as I said, awkward. It definitely could give the impression of the communication's initiator being a bit socially uncallibrated. But creepy? Are you serious? A person approaching you at 2 AM, in a dark deserted alley, laughing maniacally and trying to strike up a "casual" conversation about life after death, and then asking you whether do you think that death by bleeding is painless or not: yes, that surely would be creepy. Hearing a baby's laugh at 2 AM in your home while you are living there alone would creep me out to the point of making me insane. But some friendly and polite, even if random and clunky attempt of reaching out to fellow students? Maybe cringy at most. Come on.
We are all in a strange, unprecedented situation and there hasn't really been anything on which we could base the correct etiquette of reaching out to classmates from online course. No matter how you do it, it is inevitably going to appear more or less unusual. However, that is because the situation itself is unusual. I would expect your classmates to be able to have compassion, easily empathize with you and not deem you creepy, because they are all aware of how difficult it is to socialize in current times.
Not all people will be willing to socialize, though. But that would be independent of communication attempts: even in face-to-face settings not all people are willing to socialize, and that has nothing to do with you being "creepy". It just happens.
About potential gossip: even though it can be undesirable, it is a human thing to gossip. There are not too many interesting things happening while you are sitting home on remote learning, so anything out of ordinary is a fair game to gossip about. And people who would gossip about you, your e-mails, etc. could as well end up being great friends of yours once you get to know each other a little bit better. A few times I surely have had, post factum, some of my friends from university courses admit that they had been in fact gossiping about me before we got to know each other. Their admission was a convenient ice-breaker that allowed us to be more quick in advancing from just acquintances to actual friends who routinely tease each other in a friendly and honest banter.
I would advise using e-mail instead of Zoom. E-mail is asynchronous and wouldn't apply unnecessary pressure for people to respond at instant. In my experience, lecturers frequently and habitually send the exact same e-mails to the whole aggregated groups of students (for example, students who belong to the same group). And yes, such is modus operandi of spammers. However, real, actual spammers are people/bots who abuse the e-mail system to send messages that are completely unsolicited, contain commercial advertising, non-commercial proselytizing, phishing, attempting to scam the recipient, etc. And I am 100% sure that your e-mails to your fellow students will not contain any of those.