The runners... You might have a point with the constant force, but it's not necessarily the case. That is, if your game relies in physics, perhaps the force thing is the best solution. If not, it might be only a matter of translating your player's GameObject. You can try moving a Character Controller with a static vector, instead of getting it from input axis.
Now, for the pickups, it depends on the way you want to have it work. If those pickups are currency to buy things in your game, you just need to update a money counter each time a pickup is hit. If it's a power-up, you can activate the effect when collider is hit. Ah, those should probably work better with trigger colliders. Those might not be necessarily in a server. The server thing is a decision you have to do, depending on your needs.
For the In-App Purchases, you should have a look at Apple's documentation. Since you might not implement Apple's IAP, it'd give you enough information to get started.
