How do qubits use the "superposition" to make many trials instead of just some?

Point 1: A qubit (a particle in your words) is not rotated randomly but its rotation depends on gate acting on it. For example Hadamard gate produces a qubit with same probability of measuring $|0rangle$ and $|1rangle$ while $X$ gate is equivalent of classical NOT gate. Sequence of gates rotate qubit as intended for particular calculation. This sequence is callled quantum algorithm.

Point 2: Yes, measurement is done at the end of every algorithm to get probability distribution of a result. Based on this probability distribution you can infer a final result you are looking for. How to do this depends on particular algorithm.

Point 3: No, quantum gates are used for calculation, not for measurement.

Difference from classical calculations: Quantum computing is based on effects described by a quantum mechanics, so it is inherently probabilistic instead of deterministic. As a result, some algorithm can run faster and more efficient (e.g. Shor algorithm for integer factorization).

Superposition and trials: Superposition means that a qubit is in many different states simultaneously until it is measured. To get a probability distribution of those states you have to run algorithm many times to get enough samples for a statistics. You can repeat the algorithm as many times as you want. However, for some algorithms it is enough to run it only once to get desired result.

Last question on speed-up: The reason for speed-up comes from probabilistic nature of a quantum computer and completely different computing paradigm used in comparison with classical computers.

I would recomend reading this site: https://quantum-computing.ibm.com/support/guides/user-guide?section=5dcb2b45330e880045abccb0 I think it can help you to learn more about quantum computing.