One argument (maybe not of the kind you want) is to use the fact that the wt. 2 Eisenstein series on $\Gamma_0(p)$ has constant term (p-1)/24.
More precisely: if $\{E_i\}$ are the s.s. curves, then for each $i,j$,
the Hom space $L_{i,j} := Hom(E_i,E_j)$ is a lattice with
a quadratic form (the degree of an isogeny), and we can form the corresponding
theta series $$\Theta_{i,j} := \sum_{n = 0}^{\infty} r_n(L_{i,j})q^n,$$
where as usual $r_n(L_{i,j})$ denotes the number of elements of degree $n$.
These are wt. 2 forms on $\Gamma_0(p)$.
There is a pairing on the $\mathbb Q$-span $X$ of the $E_i$ given by $\langle E_i,E_j\rangle
= $ # $Iso(E_i,E_j),$ i.e. $$\langle E_i,E_j\rangle = 0 \text{ if } i \neq j\text{ and equals # }Aut(E_i) \text{ if }i = j,$$
and another formula for $\Theta_{i,j}$ is
$$\Theta_{i,j} := 1 + \sum_{n = 1}^{\infty} \langle T_n E_i, E_j\rangle q^n,$$
where $T_n$ is the $n$th Hecke correspondence.
Now write $x := \sum_{j} \frac{1}{\text{#}Aut(E_j)} E_j \in X$. It's easy to see
that for any fixed $i$, the value of the pairing $\langle T_n E_i,x\rangle$
is equal to $\sum_{d |n , (p,d) = 1} d$. (This is just the number of $n$-isogenies
with source $E_i,$ where the target is counted up to isomorphism.)
Now
$$\sum_{j}
\frac{1}{\text{#}Aut(E_j)} \Theta_{i,j} =
\bigg{(}\sum_{j} \frac{1}{\text{#}Aut(E_j)}\bigg{)} + \sum_{n =1}^{\infty} \langle T_n E_i, x\rangle
q^n
= \bigg{(}\sum_{j}\frac{1}{\text{#}Aut(E_j)}\bigg{)} + \sum_{n = 1}^{\infty} \bigg{(}\sum_{d | n, (p,d) = 1} d\bigg{)}q^n.$$
Now the LHS is modular of wt. 2 on $\Gamma_0(p)$, thus so is the RHS. Since we know
all its Fourier coefficients besides the constant term, and they coincide with those of the Eisenstein series, it must be the Eisenstein series.
Thus we know its constant term as well, and that gives the mass formula.
(One can replace the geometric aspects of this argument, involving s.s. curves and Hecke
correspondences, with pure group theory/automorphic forms: namely the set $\{E_i\}$ is
precisely the idele class set of the multiplicative
group $D^{\times}$, where $D$ is the quat. alg. over $\mathbb Q$ ramified at $p$ and $\infty$. This formula, writing the Eisenstein series as a sum of theta series, is then
a special case of the Seigel--Weil formula, I believe, which in general, when you pass to constant
terms, gives mass formulas of the type you asked about.)
Let me explain why Beilinson's conjecture implies that $\iota$ is the zero map (thus your first question has conditionally a negative answer).
Let $\mathcal{E}$ be a proper regular model of $E$ over $\mathbf{Z}$. The morphism $E \to \mathcal{E}$ induces a $\mathbf{Q}$-linear map $\iota : K_1(\mathcal{E})^{(2)} \to K_1(E)^{(2)}$. The image of $\iota$ is the integral subspace $K_1(E)^{(2)}_{\mathbf{Z}}$, which is also written $H^3_{\mathcal{M}/\mathbf{Z}}(E,\mathbf{Q}(2))$ in cohomological notations.
Now, what does Beilinson's conjecture predict for this group? We are concerned here with the motive $h^2(E)$, whose $L$-function is $L(H^2(E),s)=\zeta(s-1)$, and we are looking at the point $s=2$. Thus we are in the case of the "near central point" (see for example
Schneider, Introduction to the Beilinson conjectures, Section 5, Conjecture
II, or the articles by Beilinson and Nekovar mentioned in my comment).
Since the $L$-function has a pole, we have to introduce the group $N^1(E)=(\operatorname{Pic}(E)/\operatorname{Pic}^0(E)) \otimes
\mathbf{Q}$ which is isomorphic to $\mathbf{Q}$ (more generally, the dimension should be equal to the order of the pole). There is a natural injective map $\psi : N^1(E) \to H^3_{\mathcal{D}}(E_{\mathbf{R}},\mathbf{R}(2))$. Then Beilinson's conjecture asserts that $r \oplus \psi$ induces an isomorphism
$$(H^3_{\mathcal{M}/\mathbf{Z}}(E,\mathbf{Q}(2)) \otimes_{\mathbf{Q}} \mathbf{R}) \oplus (N^1(E) \otimes_{\mathbf{Q}} \mathbf{R}) \xrightarrow{\cong} H^3_{\mathcal{D}}(E_{\mathbf{R}},\mathbf{R}(2)).$$
Since the target space is $1$-dimensional and $N^1(E) \cong \mathbf{Q}$, this predicts in particular that $H^3_{\mathcal{M}/\mathbf{Z}}(E,\mathbf{Q}(2))=0$.
Moreover, it can be shown that the map $r$ is nonzero. As pointed out by profilesdroxford, the group $H^3_{\mathcal{M}}(E,\mathbf{Q}(2))$ is generated by symbols of the form $(P,\lambda)$ where $P$ is a closed point of $E$ and $\lambda \in \mathbf{Q}(P)^*$. I found a reference for this in Beilinson, Notes on absolute Hodge cohomology (Beilinson attributes this construction to Bloch and Quillen). Furthermore, in the same article the regulator of such elements is computed (in a more general setting). After some computations it turns out that $r([P,\lambda])$ is proportional to $\log | \operatorname{Nm}_{\mathbf{Q}(P)/\mathbf{Q}}(\lambda) |$. Thus $r$ is nonzero. Another useful reference is Dinakar Ramakrishnan's article on regulators.
It would be also interesting to compare the above construction with the construction proposed by profilesdroxford in his answer.
Best Answer
The standard proof (due, I believe, to Weierstrass) is as follows. The map $z\mapsto (\wp(z),\wp'(z))$ identifies $E(\mathbb{C})$ with the torus $\mathbb{C}/(\mathbb{Z}\omega _1+\mathbb{Z}\omega _2)$. Your differentials become $dz$ and $\wp(z)dz$, where $\wp$ is the Weierstrass $\wp$ function. Take for $\gamma _i$ the loop $t\mapsto a+ t\omega _i$ for some $a$. Then the periods of $dz$ are $\omega _1$ and $\omega _2$. Those of $\wp(z)dz$ are $\zeta (a+\omega _i)-\zeta (\omega _i)=\eta _i$, where $\zeta $ is the Weierstrass zeta function, a primitive of $-\wp$. The relation $\ \omega _1\eta _2-\omega _2\eta _1=2\pi i\ $ is obtained by integrating $\zeta $ along a fundamental parallelogram : see e.g. Chandrasekharan, Elliptic Functions, p. 50. The result (with a different language) appears indeed in Legendre' Traité des fonctions elliptiques, vol. I, pp. 60-61.