Is BESM a good system for this type of game?
If you like anime tropes BESM 2e Revised is my favorite for that. If you like the point buy idea of having an arbitrary value to every facet of every character, and prioritizing each character's power and talents, then I prefer dX; it's more robust, is not indicative or anime genre, and allows some great conflict resolution beyond just BFG of BFS (G=Gun, Sword=Sword; as if Big Freaking...). I absolutely love dX, but no one really plays/GMs it on here (that I've found).
Being a summoner in BESM 2e would be anything from Magic paired with Servant, to a weapon attack. As strange asa it sounds, using the attack advantages "Homing" and then your choice of the summon is great for attack summons. I have innumerous summoners that use this trick. Sure, the summons are short term, but they do that. For non-combat summons, think about what it achieves. Let's say you want the two weasels from Beastmaster (the first one) you might use Telekinesis and that represents the ferrets, whom are never actually attacked, mind you, to get the key the lock, or chew through the rope. BESM is great for reflavoring, it allows your imagination to justify the power. It, IMO, is supposed to.
dX, however, I think does it a bit better. I would be glad to talk to at length about BESM 2e Revised or dX. I'd avoid d20 BESM. If you prefered d20, it's a crapshoot, I hear some people love it. Being a D&D nut myself, one would think I'd gravitate toward d20 BESM, but quite the opposite. I haven't GMed 3e, and I prefer dX enough to not be "done" with it yet.
The shapshifter is great example of metamorphosis (IIRC--I don't have the book now) for BESM and alternate form for dX. You can build the form from scratch, as new "character" with the same identity and memories but wholly different talents and abilities; or you can just tack on "extra" stuff to your "base" form. A person who turns into a werewolf may have the second, while person who turns into a housefly will probably have the first; both which are just a different facet of spending your character points. In both cases as well, it's highly advnatageous only when you have multiple forms. You need to find the right median for the character, but one or dozen forms both end up being a hassle.
Finally, for an elemental magician, oh boy, that's easy and also complex. Anything from purely weapon attacks "fireball, ice storm, lightning strike" an array or esoteric attributes, like transmutation water to ice, or environmental control: weather, light, and temperature, could work. I wouldn't know where to start and stop with that. You could make a plethora of elementalists. Each representing their own take of it.
So yes, all these are very doable, and I personally enjoy it. It's not reductive nor restrictive. I like that as alternative to more rigid systems. I use systems so I don't have to make up stuff. I don't need to be given permission to chnage the rules; so I dislike D&Ders saying I can. Instead when I'm feeling stufled by D&D, which is occasional, but neither rare nor often, I GM or play dX.
BESM is simpler, but at the expense it has less robust conflict resoltuion; the problem with freeform games no driven by GM whim or persuasion. dX is still simple, you really only need to know what your character does, and what affects him as it does. If an enemy, for instance, has contamination, you may be keenly aware "okay, I need to figure out what this is!" but you should probably do that as it comes up.