Does anyone run into problem with their custom domain's emails being sent to other people's spam folders?

Almost everyone does.

No, although I don’t host my own email so my “reputation” is bound to my email provider (soverin.net). I’ve found https://mxtoolbox.com/SuperTool.aspx to be useful when checking email sending at work, though

Depending on how the hosting provider sets things up, you might have to create a few dns entries (spf, dkim, dmarc) to help your reputation. Also avoid using addresses historically reserved for admin stuff, like admin, mail, etc.

You need a DMARC policy, and then dkim or and spf record set up. Without these a lot of email providers will just assume your emails are spam.

https://kb.mailbox.org/en/private/custom-domains/spf-dkim-and-dmarc/

Thats not true at all

+1 for Soverin. They are a great provider. I have 4 domains and a dozen mailboxes with them.

Is there a way to do this without signing up for another subscription?

A domain? No. You'll have to pay for your domain as long as you want to keep it. They're not very expensive though. A basic .com will usually run you about 10–15 dollars per year.

Even then, a lot of email is reputation based on ip blocks. You can have all of those things correct and still be sent to spam if you’re using a cloud provider or the like. If you want to avoid trying to get up reputation up, your best bet is to use a reputable mail sending system like resend or smtp2go

I wasn’t even thinking about self hosted email, but you’re right about that.

Yes, but you'll have to use an alternative DNS system that very few people use.

You can run email on a custom domain, but as other replies demonstrate, it's a huge challenge these days. (My primary email is on a custom domain, but I probably wouldn't do it again if I hadn't already been on it for two decades.)

Yep, the big ones (MS, Google) don't give a shit about your self hosted e-mail server and throw all of your mail to spam. They don't do that with replies though.