new camera

These all sound like fairly normal difficulties when starting out with a camera so don't be too disheartened. Difficulty getting the subject (bird) in the frame is normal when you're handholding a camera with a long lens/zoom. It's a bit of an acquired skill, which just means practice! It can be good to have a few goes in the backyard or street with a slightly bigger, stationary target (one that won't hop about and fly away). It's hard to find the subject if you're waving the camera around at full zoom, and also hard to hold the camera still while you're "zooming in". With serious gear, a tripod or monopod is often used, but that's a bit overkill. Try bracing yourself or the camera against something, though, as it'll help a lot. A nearby tree, for instance, can come in really handy to to help steady everything out. More expensive cameras will have a viewfinder (eyepiece) to frame up the photo, which for most people makes it a bit easier to track a subject and keep it framed properly. Otherwise, your hands/camera move independently of your eye and it's a bit tricky to keep everything lined up. 

Your Fujifilm zooms out to a 900mm equivalent lens which is quite large. It's the same equivalent zoom as my biggest lens gives me with my DSLR (quite a common one), so it's a very usable zoom range for birds. Where possible, the preference is to get as close as possible to the bird rather than rely entirely on the zoom. Being closer should make it easier to find the bird in the camera's field of view and also give much better image quality (more detail, less blur and grain in the image). It's rarely possible to get as close as you want/need, however. The answer to "How close?" is almost always "Closer!" Sit back and let the birds themselves come closer where possible.

Compact cameras generally aren't super fast to focus and really benefit from use in good light. It should also acquire focus a bit quicker if you're not zoomed all the way to maximum, but it'll also depend on the scene/subject. (The lens lets less and less light into the camera the more you zoom in). But do use the zoom, of course! Otherwise you won't be any better off than you were with the phone.

Chopping off legs, tails and wings (and even heads and other bits) is easy to do. Avoiding this comes with experience. It's easy to get so fixated on getting the bird's head/eye where you want it in the picture that you lose track of the other bits sneaking out of the frame. Everybody does it, especially with flying birds (which are probably a bit beyond your camera's abilities). We're just a little judicial in what we choose to show. It doesn't take long before you start to gain an understanding of how "big" a bird is in the picture and you can allow a bit more space for its extremeties to also feature in the image.

Looking forward to seeing some photos. Pity about the USB cable and the batteries! Camera USB cables can be a real pain to get ahold of. Most of them have unique sockets that only work with their own brand of cable. It can be a lot less effort to buy an SD card reader which plugs directly into your computer. This might also save you some battery charge, as the camera won't need to be switched on while you're downloading pics.