1. It is not clear to me that linear duals, and not just Hodge duals, can be represented in geometric algebra at all. See, for example, here.
Can linear duals (i.e. linear functionals) be represented using the geometric algebra formalism?
2. It also seems like most tensors cannot be represented, see for example here. This makes intuitive sense since any geometric algebra is a quotient of the corresponding tensor algebra. Also it seems like only some contravariant tensors (and no covariant tensors whatsoever) can be represented unless the answer to 1. is no.
Which types of tensors admit a representation using geometric algebra?
3. The exterior algebra under which differential forms operate can clearly be represented by geometric algebra and its outer product.
However, do objects "sufficiently isomorphic" to differential forms admit a representation in geometric algebra?
4. This question seemingly depends on the answers to 1. and 3., since derivations=vector fields are the linear dual of differential forms.
Can vector fields=derivations be represented using geometric algebra?
This paper seems to suggest that the answer is yes, although it was unclear to me. It also listed as references Snygg's and Hestene's books for representing derivations=vector fields via geometric algebra. However, I quickly searched Snygg's book and could not even find the use of the word "bundle" once, which seems to cast doubt on the claim.
Moreover, derivations are just the Lie algebra of smooth functions between manifolds, correct? Since Lie algebras are non-associative, it seems doubtful to me that derivations could be represented effectively by the associative geometric algebra. On the other hand, quaternions are somehow also Lie algebras, and they can be represented in geometric algebra, so I am not sure.
5. This probably a duplicate of 4. but I am asking it anyway.
Do tangent/cotangent spaces/bundles admit a representation using geometric algebra?
This one is especially unclear to me, since using "ctrl-f" the word "bundle" is not used even once in Snygg's book "Differential Geometry via Geometric Algebra", which appears to be the most thorough treatment of the subject.
(Incidentally, the word "dual" also only appears once, in reference to Pyotr Kapitza's dual British and Russian citizenship.)
Basically I am wondering if differential geometry can be "translated" completely using the language of geometric algebra. I think the answer is no because Hestene's conjecture regarding smooth and vector manifolds has yet to be proved (see the comments here), but it seems like we would run up with barriers even sooner than that. Although I probably am misunderstanding the comment.
I have found differential geometry difficult to understand at times, and would like to learn it by translating it as much into geometric algebra and then back. The extent to which the two is "equivalent" obviously presents a barrier to how much this is possible. Still, I already feel like I understand the concepts and motivations of multilinear algebra and related fields much better after having just learned a little geometric algebra, and would like to apply this as much as possible to the rest of differential geometry.
These questions are also related: symmetric products are the inner product from geometric algebra, and wedge products are the outer product from geometric algebra; geometric algebra is a special type of Clifford algebra which contains the exterior algebra over the reals; and this question discusses derivations in algebras in detail.