In this paper, we consider the problems from the area of sublinear-time algorithms of edge sampling, edge counting, and triangle counting. Part of our contribution is that we consider three different settings, differing in the way in which one may access the neighborhood of a given vertex. In previous work, people have considered indexed neighbor access, with a query returning the i-th neighbor of a given vertex. Full neighborhood access model, which has a query that returns the entire neighborhood at a unit cost, has recently been considered in the applied community. Between these, we propose hash-ordered neighbor access, inspired by coordinated sampling, where we have a global fully random hash function, and can access neighbors in order of their hash values, paying a constant for each accessed neighbor. For edge sampling and counting, our new lower bounds are in the most powerful full neighborhood access model. We provide matching upper bounds in the weaker hash-ordered neighbor access model. Our new faster algorithms can be provably implemented efficiently on massive graphs in external memory and with the current APIs for, e.g., Twitter or Wikipedia. For triangle counting, we provide a separation: a better upper bound with full neighborhood access than the known lower bounds with indexed neighbor access. The technical core of our paper is our edge-sampling algorithm on which the other results depend. We now describe our results on the classic problems of edge and triangle counting. We give an algorithm that uses hash-ordered neighbor access to approximately count edges in time Õ(n/"m + 1/"2) (compare to the state of the art without hash-ordered neighbor access of Õ(n/"2 m) by Eden, Ron, and Seshadhri [ICALP 2017]). We present an ω(n/"m) lower bound for "≥m/n in the full neighborhood access model. This improves the lower bound of ω(n/s"m) by Goldreich and Ron [Rand. Struct. Alg. 2008]) and it matches our new upper bound for "≥ m/n. We also show an algorithm that uses the more standard assumption of pair queries ("are the vertices u and v adjacent?"), with time complexity of Õ(n/"m + 1/"4). This matches our lower bound for "≥ m1/6/n1/3. Finally, we focus on triangle counting. For this, we use the full power of the full neighbor access. In the indexed neighbor model, an algorithm that makes Õ(n/"10/3 T1/3 + min(m,m3/2/"3 T)) queries for T being the number of triangles, is known and this is known to be the best possible up to the dependency on "(Eden, Levi, Ron, and Seshadhri [FOCS 2015]). We improve this significantly to Õ(min(n,n/"T1/3 + n m/"2 T)) full neighbor accesses, thus showing that the full neighbor access is fundamentally stronger for triangle counting than the weaker indexed neighbor model. We also give a lower bound, showing that this is the best possible with full neighborhood access, in terms of n,m,T.