Your Nikon D3300 (and pretty much every other film SLR & DSLR) meters and autofocuses with the aperture of whatever lens is mounted on it wide open at maximum aperture. It doesn't stop the lens down to the selected aperture setting until the instant before the shutter opens to expose the image. What aperture setting is selected is immaterial in terms of AF performance - it's the lens' maximum aperture that counts.
Here is the way the crop factor affects autofocus in terms of the lens' aperture and whether the lens is a DX or FX lens: it doesn't.
A DX lens at f/6.3 is the same as an FX lens at f/6.3 when both are used on an APS-C camera, just as a 300mm DX lens and a 300mm FX lens are the same focal length and provide the same field of view when both are mounted on an APS-C camera.
The narrower baseline allowed by the smaller mirror of an APS-C camera will mean some reduction in the maximum theoretical AF performance when compared to a FF camera with the same generation of AF technology. This is because the width of the baseline from one side of the system to the other can affect the speed and accuracy of a PDAF system. But that is true no matter what lens you have mounted on any APS-C camera. In the case of Nikon, they seem to concentrate more on making their AF systems work with as many lenses as possible at the expense of giving up the superior performance for which they might be capable using very wide aperture lenses.
The only AF issues associated with this lens and Nikon DSLRs that I've been able to find widespread reports of are only associated with the very early examples of the lens when it was first introduced around 2010 and then later issues with some subsequent models of Nikon cameras (particularly the D5300) that required a firmware upgrade to make the lens compatible with the newer camera. The first issue was resolved and copies of the lens made since then have not had that issue. The second issue can potentially occur anytime a new camera model is introduced. It's just part of the risk one takes when buying a reverse engineered third party lens. Sigma has usually been very good about updating firmware for current and recent lenses when incompatibilities with newer camera models are discovered. But eventually discontinued lenses stop getting updates for the latest cameras. Since your D3300 was introduced while the lens in question was still in production that shouldn't be much of an issue for you.
So as far as the f/6.3 maximum aperture at focal lengths of 313mm and above, there don't seem to be any widespread reports that Nikon cameras won't AF with the lens. Of course any camera should AF better (faster, more accurate, more consistent from shot-to-shot) with a wider aperture lens than a narrower aperture lens all else being equal. It is also true that any camera/lens combination will AF better in brighter light than in dimmer light and that high contrast targets are easier to AF than low contrast targets. One shouldn't expect an f/5.6 lens to perform as well with regard to AF as an f/2.8 lens will, even when both are stopped down to the same aperture when the picture is actually taken. As mentioned above, they all meter and AF with the aperture at its maximum setting. Often when manufacturers say their AF system is good to f/5.6 what they really mean is anything wider than f/8.
There are, however, some important considerations one should weigh when entertaining the possibility of using the Sigma 150-500mm f/5-6.3 DG OS HSM APO:
- The lens has been replaced with a newer model, the Sigma 150-600mm f/5-6.3 DG OS HSM Contemporary. The newer lens has better optical performance, particularly at the longer end of the focal range where the lens will probably tend to be used the most. It also has the ability to have firmware updates downloaded and installed by the end user using an inexpensive USB dock rather than having to ship the lens to a Sigma service center to have any needed updates installed as would be the case with the older 150-500mm.
- No matter what lens and camera body one is using, 500mm focal length requires excellent shooting technique. Rare is even the seasoned pro that can shoot at 500mm handheld and not get some blur from camera movement. Yes, optical stabilization helps in this regard. But OS/VR/IS/etc. isn't perfect or foolproof at such long focal lengths. The APS-C crop factor will exacerbate the issue, as you are only capturing a very narrow FoV equal to a 750mm lens on a 35mm/FF camera. So plan on using a solid tripod with a very good head that can handle the weight of such a large lens, or at the very minimum use a good monopod when shooting with this lens at the longer focal lengths.
- Lower priced telephoto zooms in this price range tend to get softer at the long end of their focal range. Out past about 350-400mm the sharpness of the Sigma 150-500mm drops off noticeably. That's not to say you can't get some good shots with it. But you can't enlarge images as much when displaying them before the imperfections begin to show. Thom Hogan's review of this lens illustrates this very well. The newer Sigma 150-600mm Contemporary does a lot better in this regard, but it is still at its softest at the longest focal length. That's the price one pays for relatively inexpensive long focal length zoom lenses.
- Your Nikon D3300 does not have AF Fine Tune capability. For longer focal length lenses this capability is a near necessity. The newer Sigma 150-600mm gives the ability to calibrate AF using the USB dock mentioned above. The older Sigma 150-500mm doesn't. If the combination of your particular camera and your particular lens has a consistent front or back focusing issue there's nothing you will be able to do about it short of sending both your lens and your camera to Sigma for calibration to each other.
As much as with any other type of lenses, when one is talking about long focal length telephoto lenses it really is true that you get what you pay for. If you can afford the older Sigma 150-500mm f/5-6.3 but can't quite swing the cost of the newer Sigma or Tamron 150-600mm f/5-6.3 offerings then perhaps that is a good option for you. But if you can save a bit longer and get either the Sigma 150-600mm f/5-6.3 Contemporary or the Tamron SP 150-600mm F/5-6.3 Di VC USD instead you'll probably be much happier with the results you can get with either one of those lenses.
You can find some very good reviews and comparisons of all of these lenses at The-Digital-Picture. At the end of the review for the Sigma 150-600mm Contemporary that lens is compared to both the Sigma 150-500mm and the Tamron 150-600mm as well as a couple of other lenses.
Personal note: I've got a friend that had a Sigma 150-500mm f/5.6-6.3 DG OS HSM in the Canon EOS mount back in around 2012. It wasn't as sharp past 350mm as he wanted and he sold it after only having it about 6 months or so as he could just crop images from a quality 70-300mm and get results that were just as good.