Are products such as this suitable as an external oscilloscope for a eurorack system?

Hantek 2C42 Handheld Oscilloscope Multimeter 2 in 1 Multifunction Tester 2CH+DMM 40MHz Scope: https://www.amazon.com/Hantek-Handheld-Oscilloscope-Multimeter-Multifunction/dp/B07PWZV4XJ

Absolutely, yes. O'scopes can handle a couple of hundred volts, depending on the model.

I'm really unclear on if I can run a patch cable directly from one of my modules into this device. I'm also not sure if this oscilloscope works with DC signals. It says the multimeter is DC, but nothing about the oscilloscope.

Because it's assumed that you know that the scope handles both DC and AC.

Tangentially: I have no multimeter and I'm thinking that would be useful for testing hardware for issues, recalibrations, and building DIY modules. But I see I can get a dedicated multimeter for a tenth the price.

And you're absolutely right...a multimeter IS more important than an oscilloscope. The only practical uses I see for a scope is for checking waveform purity, working on a module, and hunting noise in signals, plus they're quite indispensable for checking phasing when mixing. But a multimeter works on so much more that's germaine to electronic music...from troubleshooting the latest eBay find to precise CV settings, especially in harmonic additive methods that require exacting CV manipulation, to finding levels from experimental devices (like test gear!) that are appropriate for synth CVs or audio signals. Over on Hainbach's subreddit, one constantly-repeated point is that you NEED a multimeter if you're doing hardware-based electronic music. And that ain't no joke!

-- adamj