Regarding Bulb Mode
If you use a wired remote there is generally not a time limit regarding the length of an exposure using bulb mode with most current DSLRs. Pressing the button, halfway or fully, on a wired remote is pretty much identical to pressing the camera's shutter button (except you don't physically touch the camera).
There have been some DSLRs (e.g. the Nikon D50) in the past that limited maximum bulb exposures to 30 minutes. There might still be some models with such limitations.
If you use a wireless infrared remote there are some camera systems that limit the maximum bulb exposure length to 30 minutes or less. Nikon cameras using time mode with an ML-L3 wireless remote have such a limitation. Most infrared remotes also don't allow a half press of the shutter button.
Wireless infrared emotes also usually limit you to a single exposure per shutter press. Though this wouldn't affect a single shot in bulb mode, you might want to shoot continuous frames to use for image stacking. Again, wired remotes have no such restrictions.
In this answer to Can I use a remote shutter / bulb mode on a Canon T4i? I discuss the merits of a wired shutter release versus a wireless one. I've got both an infrared remote and wired cable releases. One of my wired releases has an intervalometer built in, the others just have a single button that can do a half press, full press, and include a slide lock that can lock the button down in a full press. I use the wired releases most all of the time when I need a remote release. I use the wireless infrared remote about once every year or two.
If you choose to use the camera's self timer, many of them will not allow an exposure longer than 30 seconds. The Nikon D3xx0 and D5xx0 series (i.e. D3300, D5500) have such a restriction.
Dark Frames: Many cameras do what is called dark frame subtraction with longer exposures. They do this to reduce the effect of heat buildup in the sensor during long exposures that can cause hot pixels. The camera basically takes another image with the shutter remaining closed immediately after the exposure. This dark frame is the same length of time as the original exposure. The camera then subtracts the noise in the dark frame from the original exposure. This means if you take a ten minute exposure you then must wait another ten minutes for the camera to take the dark frame. Most DSLRs will allow the user to turn it off or on. Canon calls it Long Exposure Noise Reduction.
Some older Pentax cameras, however, do not allow the user to turn this feature off. When doing image stacking, which combines several multiple images, this can be a real headache. Most image stacking programs will let the user periodically take a few dark frames manually during a session and insert them so the program can subtract the dark frames from the other frames. Not having to spend half the session waiting on the camera to do an automated dark frame allows the work to move much faster.
Mirror lockup: DSLRs have a reflex mirror that cycles up and down each time an image is taken (unless the camera is Live View mode). Normally mirror lockup is only needed for shutter times of around 1/100 second to about 1 second. The influence of the vibrations from the mirror movement are normally reduced for exposures longer than one second. Astrophotography, though, can be the exception that proves the rule. That is because most of the field of view is very dark with only a few points of very bright lights. Even with longer exposures, the vibrations of the mirror can cause short star trails in the otherwise very dark areas around each star. Some entry level DSLRs don't have the option of mirror lockup. You can work around this limitation by using Live View, but that can create other issues when doing astro work.
Noise Reduction: Some cameras are known as "star eaters" because the way they do noise reduction on the sensor can eliminate dimmer stars as random noise. Nikon models with Hot Pixel Suppression (HPS) have been called 'star eaters' by some astrophotography enthusiasts. Ditto, for different reasons involving only bulb exposures, for most of Sony's recent A7 series of cameras. There's current debate going on, based on observations from pre-production testers, about whether the new yet-to-be released in the wild Sony A7 R III is a star eater or is not a star eater. For a long period many astro enthusiasts preferred Canon DSLRs because they did not do this type of NR on the sensor itself. Since the introduction of a new sensor in the EOS 80D in 2016, though, Canon appears to have moved in the same direction as what Sony and Nikon have been doing for quite a few years with regard to on-die noise reduction.
Image stacking: We've got more than a few questions here that deal with image stacking. They're a great resource for learning how to get nice images of the night sky. It's a technique used to average out the random noise in images taken of the night sky. Most of the really impressive amateur astrophotography images you've seen were very likely produced using image stacking. So do many astro photos professionally produced by observatories. Cameras that allow you to automate a large number of images shot in sequence will make doing image stacking a lot easier.
Special "astronomy" camera models: Some camera manufacturers have introduced special "astrophotography" models at various times. Canon's latest was the EOS 60Da which was a modified 60D set up for astro work. Sadly, the 60Da is no longer available. Nikon's D810a is still listed as a current model, but it seems to be on backorder more than it is available at major online camera sellers such as B&H. And as a specialized version of a fairly expensive FF camera, it isn't exactly cheap, either. Such models usually alter the filters in front of the imaging sensor to allow more of non-visible wavelengths of light (e.g. Hα) to make it to the sensor and be recorded. This allows objects, such as nebulae, that radiate primarily in Hydrogen-α to be better photographed.
There is also a rich selection of questions here with the astrophotography tag. You might find many of them helpful!