Is this normal for RAW images?
A RAW image is just that: raw. It is as close to the exact output of the sensor as you can get out of a given body. Critically, in most cases it will not even have been debayered, which gives you the opportunity to perform this step yourself (which can sometimes result in better image recovery compared to the on-body algorithm, which is almost always going to make some quality-for-speed trade-off), and better post-processing applications will allow you to experiment with different algorithms.
Consequently, of course, this means that the pixels will start with a one-to-one correspondence to sensor elements, which means no distortion correction. (Note that this is technically more accurate to film, which also would not have distortion correction.) As you probably know, the camera body does considerable post-processing of this RAW image in order to produce a JPEG, which may include operations such as noise reduction, correction for chromatic aberration, and distortion correction.
What is the best way to deal with it? Do I just have to eyeball it to figure out how much lens correction I have to use in lightroom?
Like debayering, better software will have options for correcting (or applying!) distortion. As a firm believer in Free(dom) Software I'm most familiar with lensfun (which is incorporated into e.g. GIMP and darktable). Such tools typically operate using a database of known lenses (I believe the only relevant property of the body is the sensor size). Some may include mechanisms to calibrate an unknown lens, which typically requires "reference images" containing straight lines (e.g. photographing a test chart). Basically, if you're using a "known" lens, this should just work "automagically", but (at least if you use Free software) if you have an "unknown" lens, you still have options if you're willing to get your hands dirty.
Short version: I can't speak to Lightroom as I morally avoid proprietary software, but darktable+lensfun should apply distortion correction automatically for known lenses. (There might be a box you need to tick, but you shouldn't need to futz with the distortion parameters.)
If you have an unknown lens and you don't want to go to the effort of actually calibrating it, another option would be to take a photograph of a test chart using the same lens and focal length, and use that to "eyeball" the necessary correction. You can then apply the same correction to your "real" image.
Will the amount of distortion be exactly the same for every RAW image I will take with that camera + lens combo? Or does it depend on focal length, exposure parameters and other factors that might differ per image?
AFAIK, only focal length matters. (Focal distance and aperture might, but exposure certainly does not. At any rate, most correction algorithms only care about focal length. And of course what lens you are using, but you asked about "for a particular lens". In fact, using a different body shouldn't matter as long as it uses the same sensor and mount, but a full-frame vs. DX sensor will matter. Hopefully, any body will have the lens' principle point centered on the sensor and the sensor parallel to the lens' focal plane, i.e. the mount ought to be universal.)
Keep in mind, distortion is a property of the camera optics and how they affect the path that light takes between the scene and the sensor. The shape of the lens assembly will obviously affect this, which includes focal length for variable zoom lenses. How long you expose the sensor obviously won't affect this. Other factors, such as aperture, focal distance and the distance between the camera and the subject (i.e. what you're shooting) may have an impact, but I believe it is significantly less than the "general" distortion and is typically ignored.
(One other note: software almost always assumes that the principle point is the center of the image. This is partly because this is almost always a safe assumption, but keep in mind it means you have to apply distortion correction before cropping an image.)