Yes, chucks are purchased separately from the lathe. Be sure the chuck thread size matches the headstock threads. (Less common: Some chucks use adapters, where you buy a chuck and also the adapter for it that matches your headstock threads.)
The most common headstock thread size is 1 inch x 8 TPI. The benchtop lathes you are considering should all have 1 inch x 8 TPI headstock threads. (Larger lathes may come with 1-1/4 inch x 8 TPI.)
Lathes also have tapers on both ends. The tapers are used for other accessories like drive spurs and live centers for spindle turning. The most common taper size is Morse Taper 2, also known as MT2. The benchtop lathes you are considering should all have MT2 tapers. (You may find one with MT1 tapers. That's okay, but not the best choice.)
Please do your homework regarding turning tools.
High speed steel (HSS) turning tools come in different types. Common HSS turning tools include bowl gouges, roughing gouges, spindle gouges, skew chisels, parting tools, and various sizes and shapes of scrapers. What they all have in common is that they need frequent sharpening. Woodturners often stop turning to give their tool a quick touch up on the sharpener. If you buy HSS turning tools, you MUST have a sharpener for them. They may not be sharp from the factory, and won't stay sharp for long, especially for a beginner.
I recommend a low speed grinder and the Wolverine jigs. (I get by with a Grizzly wet grinder and Tormek jigs, but wish I had the low speed grinder and Wolverine jigs instead.)
The other choice is carbide. Carbide turning tools use replaceable carbide tips which last a long time. The manufacturers expect you to replace the tips when they get dull, but you can touch them up on diamond stones. The handle tips and the carbide cutter shapes are specific to each other - you can't attach a round cutter to a square-radius handle tip. Popular carbide turning tool cutter shapes are round, square-radius, square, and diamond. There are others. I recommend square-radius over square. Carbide turning tools are all considered "scrapers."
Some people have strong opinions about HSS vs. carbide. I do not. I use both kinds of tools and like both kinds of tools. Both offer their own "fun factors", and it is the fun that counts, right?