I am looking for references of the following two questions:
1) For with class of primes the rank of the elliptic curves $y^2=x^3+px$ is exactly $0,1$ or $2$. It was quite easy to show that the rank is $0$ for $p\equiv 7$ or $11 \pmod {16}$. But what about other primes? How "good" we know the tank these days?
2) Is there any method (which an undergrad student could understand) for determining generators of the group $E(\mathbb Q)$ when $rank(E(\mathbb Q))>0$ and $E$ is given by $y^2=x^3+ax$?
In Silverman's book "The Arithmetic of Elliptic Curves" the classical results on the Mordell-Weil rank of these curves are summarised in chapter $X$, section $6$, which is called "The Curve $E: Y^2=X^3+DX$" for fourth-power free integers $D$. Proposition $6.2$ gives us the formula, for $D=p$ prime, $$ {\rm rank} \; E(\mathbb{Q})+\dim_2 Ш (E/\mathbb{Q})[2]=\begin{cases} 0 \text{ if } p\equiv 7,11 \bmod 16 \\ 1 \text{ if } p\equiv 3,5,13,15 \bmod 16 \\ 2 \text{ if } p\equiv 1,9 \bmod 16 \end{cases} $$ The left-hand side is called the Selmer rank. From this we can derive that for $p\equiv 7,11 \bmod 16$ the rank of $E(\mathbb{Q})$ is $0$, and $Ш (E/\mathbb{Q})[2]=0$. For $p\equiv 5\bmod 8$ the rank is at most $1$; it would follow from the Birch-Swinnerton Dyer conjecture that it has rank exactly $1$ for all primes $p\equiv 5\bmod 8$, but this is of course open. So taking together all remarks in this section of Silverman's book you will obtain an answer, what was known until, say, $1992$, about primes $p$ with rank $0$, $1$, or $2$. I think the experts can tell you what the status is today. There are of course several papers on the web, which you might want to look at, e.g., Goto's paper from $2001$, and the references, and his PhD thesis.