Confusion about quantum field in AQFT
Actually, unless dealing with Weyl operators, dealing with bosonic fields on a given spacetime $M$, the field operators are elements of a $*-$ algebra with unit $\cal A$. Abstract boson field operators are algebra-valued linear functions $$C_0^\infty(M) \ni f \mapsto \phi(f) \in \cal A \:,$$ satisfying some further properties. The algebra is made of finite linear combinations of the unit and finite products of these elements, smeared with different functions $f$. One speaks of distributions when further topological hypotheses are assumed about the above functions with respect to some semi-normed topologies usually induced by algebraic states (positive linear functionals on the algebra). You see that, at this stage of the formalism, no product of distributions enters the game, since the products are taken between fields smeared with different test functions: we have only something like $$\phi(f)\phi(g) = ``\int_{M\times M} \phi(x) f(x) \phi(y) g(y) d\mu(x) d\mu(y)"$$ and not bad objects like this $$\phi^2(f) = ``\int_{M} \phi(x) \phi(x) f(x) d\mu(x)"\:. \tag{1}$$
For the scalar boson field, a proper $C^*$ algebra can be introduced referring to formal objects $e^{i\phi(f)}$ called abstract Weyl operators. The problem is that the use of this $C^*$ algebra turns out quite cumbersome (if not impossible) when dealing with (self-)intracting models. However (nets of) $C^*$ algebras are found, even starting from simple $*$-algebras, when passing to a Hilbert space formulation after having fixed an algebraic state. These special $C^*$-algebras are in fact von Neumann algebras.
The observables of the theory are the formally selfadjoint elements of the algebra though, barring the case where one deals from scratch with proper unital $C^*$ algebras, this is a quite naive interpretation and many problems affect it when passing to the representation of the algebra in a Hilbert space associated to a given algebraic state through the so-called GNS procedure. Elements of the $*$-algebra which are selfadjoint in the algebra are not (essentially) selfadjoint in the Hilbert space. They may admit none or many selfadjoint extensions and this is an overlooked open problem in AQFT formulated with $*$-algebras, though there exist some recent partial results.
Conversely, the algebraic approach is quite effective in describing notions a bit vague or cumbersome as the spontaneous breaking of symmetry and also to simultaneously handle unitarily inequivalent representations of the same algebra of observables. Also the thermodynamical limit becomes easy: no limit at all! The plethora of algebraic states is so large to include states which already describe the thermodynamic limit from scratch. They are not density matrices and thermality is encoded in the KMS condition. The AQFT language is very natural to handle QFT in the presence of black holes, to describe the Hawking radiation and the Unruh effect. Also the interpretation of the Hawking effect as a tunnel effect has a natural and rigorous version in AQFT in terms of properties of Hadamard states. (As a personal note I might say that I spent around 20 years of my career on these subjects, now I am passing to other stuff.)
The said unital $*$ algebra $\cal A$ of a (bosonic) AQFT is however too small to include all interesting observables of the theory as, for instance, the stress energy tensor whose expectation value is seen as the source of the gravitational field in a semiclassical formulation of quantum gravity. The most elementary versions of these objects are exactly of the form (1). Enlargements of the algebra contain in fact suitable generalizations of products of distributions and there (1) makes sense. The fact that these products are ill-defined is the source of (finite) ultraviolet renormalization.
The ultraviolet renormalization procedure is described in the AQFT in a direct way immediately achieving the finite renormalization counterterms without passing throught infinite quantities.
All that refers to AQFT in a given spacetime. It is possible to formulate a more advanced AQFT that simultaneously considers all possible globally hyperbolic spacetimes using the language of the theory of categories. Here a bit vague notions as general covariance are encoded in the precise functorial language. This approach has some interesting consequences, for instance it implies that the values of the renormalization constants are the same for all spacetimes.
This paper I have authored with I. Khavkine should be a sufficiently smooth introduction to these ideas.
That is part (chapter 5) of a book collecting some recent results on the subject including the categorial approach and the renormalization procedure.
The two parts of your question illustrate two of the motivations for the AQFT approach: to focus on observables instead of on fields, and to use only bounded operators — ordinary operators that are well-defined on the whole Hilbert space.
In a canonical formulation of QFT, field operators are the fundamental objects (mathematically), and observables are expressed in terms of them. In a typical model, most combinations of field operators are not observables. In contrast, AQFT is based directly on observables. In general, the $C*$-algebras in AQFT don't include field operators, except in special models where (some of) the field operators can be expressed in terms of observables. Fields can still enter indirectly, though, by considering relationships between different Hilbert-space representations of the same net of observables.
One of the conceptual advantages of the AQFT approach (not a computational advantage, but a conceptual one) is to avoid distributional "operators," instead using only operators that are truly well-defined on the whole Hilbert space. That's why AQFT talks about the algebras associated with topologically open subsets of spacetime, instead of talking about operators associated with points of spacetime.
Both of these issues can be understood relatively easily using lattice AQFT,$^\dagger$ which is just an AQFT-like approach to lattice QFT. For example, here's how to apply it to question 2: Operators associated with points in lattice QFT can be well-defined operators on the whole Hilbert space, but when we consider the continuum limit of the canonical commutation relations, we immediatley see that an operator can remain bounded only if it is suitably "smeared" over some region that remains finite (not pointlike) in the continuum limit.
$^\dagger$ I've never seen the name lattice AQFT used before. I just pulled it out of the air while writing this answer, because it needs a good name. It might not help us calculate much, but it can still help clarify some things conceptually. In a culture dominated by perturbation theory, having a conceptually clear and mathematically unambiguous non-perturbative way of thinking about things can be worth a lot. And, unlike AQFT in continuous spacetime, lattice AQFT is known to accommodate many non-trivial and physically-relevant models, including (lattice) QED and (lattice) QCD.