Group Theory – Do Generators Belong to the Lie Group or the Lie Algebra?

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In Physics papers, would it be correct to say that when there is mention of generators, they really mean the generators of the Lie algebra rather than generators of the Lie group? For example I've seen sources that say that the $SU(N)$ group has $N^2-1$ generators, but actually these are generators for the Lie algebra aren't they?

Is this also true for representations? When we say a field is in the adjoint rep, does this typically mean the adjoint rep of the algebra rather than of the gauge group?

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If you have a basis for the Lie algebra, you can talk of these basis vectors as being "generators for the Lie group". This is true in the sense that, by using the exponential map on linear combinations of them, you generate (at least locally) a copy of the Lie group. So they're sort of "primitive infinitesimal elements" that you can use to build the local structure of the Lie group from.

Re your second point, yes, fields in gauge theories are generally Lie algebra-valued entities.

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[Physics] Weinberg’s way of deriving Lie algebra related to a Lie group

Here is how I understand Weinberg's discussion:

First of all: following Weinberg's words precisely, he only says that "Such a set of commutation relations is known as a Lie algebra". He motivates the commutation relations by considering a representation $U(T)$ of a Lie group $T$, but he does not clarify the relationship between these notions. The commonly accepted terminology has to be learned from elsewhere; I'll try to expand on it here.

Note that mathematicians distinguish between "a Lie algebra" and "the Lie algebra associated to a Lie group".

Any algebra of operators that fulfills (2.2.22) is "a" Lie algebra.

But for each Lie group $T$, there is "the" "most general" Lie algebra $\mathfrak{t}$ that fulfills (2.2.22). It arises from the structure constants $f^a_{bc}$ that Weinberg mentions. Note that the structure constants do not depend on the representation on some Hilbert space, the only depend on the Lie group $T$.

To find this "most general" Lie algebra associated to a Lie group, you can look at a very special vector space, namely the tangent space at the identity element $1\in T$. The ajdoint representation of the group on this space will give rise to a Lie bracket $[·,·]$ on that vector space and turn it into said "most general" Lie algebra.

As for your particular questions:

Weinberg only the mentions "a Lie algebra", though it comes from a group. It's "the" lie algebra of the group $U(T)$.
In general, every such Lie algebra is a homomorphic image of "the" Lie algebra associated to $T$. These Lie algebras are different from each other; for instance, the vector spaces may have different dimensions.
Weinberg invokes Wigners theorem for the following reason: a priori, the symmetry group $T$ acts on rays. Remember that the vector $|ψ\rangle$ represents the same physical state as the vector $ξ|ψ\rangle$ which arise from multiplication with an arbitrary complex number of mangitude $|ξ|=1$. A ray is the set of all vectors that arise in this fashion; they all represent the same physical state.

Now, the symmetry maps physical states to physical states, i.e. sets of vectors to sets of vectors. But each symmetry operation is allowed to permutate vectors within the set, after all, they are indistinguishable physically. It is not clear at all that a mapping $T_1$ on rays can be recast as a mapping $U(T_1)$ that acts on individual vectors and is linear. Neither is it clear that $U(T_1T_2) = U(T_1)U(T_2)$, because these operations may map the rays to each other, but they might permute the vectors within a ray rather differently, which would manifest in a phase $U(T_1T_2)|ψ\rangle = ξ·U(T_1)U(T_2)|ψ\rangle$.

We want the symmetry to act on a Hilbert space, though, because this allows us use the familiar expression for the commutator of two operators $[X,Y] = XY-YX$. After all, it involves the addition (subtraction) of two operators, something that is not available for symmetries merely acting on rays.

Other than that, Hilbert spaces are completely unnecessary for defining Lie algebras or Lie groups.

The question of how to find all representations of a Lie group or of a Lie algebra is beyond the scope of this answer. It leads to topics such as semisimple Lie algebras and their classification.
Mathematicians obtain the Lie algebra associated to a Lie group by considering the tangent space at the identity element, as mentioned above. The Lie group doesn't need to be connected for this to be well-defined. (But it is only useful for studying the part of the group that is connected to the identity.)

[Physics] Trace of generators of Lie group

More generally, let there be given an $n$-dimensional complex Lie algebra $(L, [\cdot,\cdot])$.

The adjoint representation ${\rm ad}:L\to {\rm End}(L)$ is defined as $$({\rm ad}x)y~:=~[x,y], \qquad x,y~\in~L.\tag{1}$$
The Killing form $\kappa:L\times L\to \mathbb{C}$ is defined as $$ \kappa(x,y)~:={\rm tr}({\rm ad}x \circ {\rm ad}y), \qquad x,y~\in~L, \tag{2}$$ is bilinear, symmetric, associative. It is non-degenerate iff $L$ is semisimple.
One may show that any bilinear, symmetric, associative form $L\times L\to \mathbb{C}$ is proportional to the Killing form $\kappa$ if $L$ is simple.
Given an arbitrary basis $(t_a)_{a=1,\ldots, n}$ for $L$, define the structure constants $f_{ab}{}^c\in\mathbb{C}$ via $$ [t_a,t_b]~=~\sum_{c=1}^nf_{ab}{}^ct_c , \qquad a,b~\in~\{1,\ldots, n\}. \tag{3}$$
Define Killing metric $$ \kappa_{ab}~:=~\kappa(t_a,t_b) , \qquad a,b~\in~\{1,\ldots, n\}.\tag{4}$$
One may show that the lowered structure constants $$ f_{abc}~:=~\sum_{d=1}^nf_{ab}{}^d\kappa_{dc}\, \qquad a,b,c~\in~\{1,\ldots, n\},\tag{5}$$ are always totally antisymmetric in the indices $abc$.
It is possible to choose the basis $(t_a)_{a=1,\ldots, n}$ such that the Killing metric $\kappa_{ab}$ is diagonal; and in the semisimple case, such that $\kappa_{ab}$ is proportional to the identity $\delta_{ab}$.

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[Physics] Weinberg’s way of deriving Lie algebra related to a Lie group

[Physics] Trace of generators of Lie group

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