I have several questions as the title says, so I will try to explain how doubts appeared by following the book "Mathematical Methods in Quantum Mechanics" written by Gerald Teschl, precisely chapter 1.4.
At first, it is defined $$\mathcal{F}(H,\tilde{H})=\{\sum_{j=1}^n\alpha_j(\varphi_j,\tilde{\varphi}_j)\mid (\varphi_j,\tilde{\varphi}_j)\in(H,\tilde{H}),\alpha_j\in\mathbb{C}\},$$ where appear my first question:
1.- At this point we must think on $\alpha(\varphi,\tilde{\varphi})$ as $(\alpha\varphi,\alpha\tilde{\varphi})$ or more like a triplet $(\alpha,\varphi,\tilde{\varphi})$?
It follows the definition of $$\mathcal{N}(H,\tilde{H})=\text{span}\{\sum_{j,k=1}^n\alpha_j\beta_k(\varphi_j,\tilde{\varphi}_k)-(\sum_{j=1}^n\alpha_j\varphi_j,\sum_{k=1}^n\beta_k\tilde{\varphi}_k)\},$$ and of course we are going to consider $\mathcal{F}(H,\tilde{H})/\mathcal{N}(H,\tilde{H})$, because as said in the book it is in this set of classes where we have the next three properties:
a) $(\varphi_1+\varphi_2)\otimes\tilde{\varphi}=\varphi_1\otimes\tilde{\varphi}+\varphi_2\otimes\tilde{\varphi}$.
b) (the same for the right)
c) $(\alpha\varphi)\otimes\tilde{\varphi}=\varphi\otimes(\alpha\tilde{\varphi})$.
On $\mathcal{F}(H,\tilde{H})/\mathcal{N}(H,\tilde{H})$ it is defined the inner product $$\langle\varphi\otimes\tilde{\varphi},\psi\otimes\tilde{\psi}\rangle=\langle\varphi,\psi\rangle\langle\tilde{\varphi},\tilde{\psi}\rangle,$$ and then the tensor product is defined as the completition of $\mathcal{F}(H,\tilde{H})/\mathcal{N}(H,\tilde{H})$ with this inner product.
From here, my questions are about the examples and the proposed problems:
2.- Why it is easy to see that $H\otimes\mathbb{C}^n=H^n$?.
3.- Show that $\varphi\otimes\tilde{\varphi}=0$ if and only if $\varphi=0$ or $\tilde{\varphi}=0$.
4.- Show that $\varphi\otimes\tilde{\varphi}=\psi\otimes\tilde{\psi}\neq 0$ if and only if there is some $\alpha\in\mathbb{C}\backslash\{0\}$ such that $\varphi=\alpha\psi$ or $\tilde{\varphi}=\alpha^{-1}\psi$.
Remark: Problems 3 and 4 are easy from right to left by using the three properties mentioned before, but in the other direction I have just tried to use the definition of the tensor product, but writing the elements of the tensor product as cosets is quite cumbersome and doesn't take me anywhere.
Any help would be really appreciated even for the first question, that I think it must be just a misunderstanding.
Regarding your first question: $\mathcal F(H,\bar H)$ is simply the set of all finite (complex) linear combinations of elements of $H\times\bar H$, the cartesian product of the two Hilbert spaces. Maybe you are confused because you put the index in the wrong place but $(\varphi_j,\bar\varphi_j)$ is nothing but a vector.
The second question is just, as Teschl says, taking a basis for $\mathbb C^n$. If it helps, take his equation (1.43) and identify $H\otimes\mathbb C$ with $H$.