With a new fill valve, the slower filling could be a result of either:
- a different fill valve itself
- blockage in or a failing cutoff valve outside the toilet
- debris in the fill valve from not purging water through the valve during the install
New styles may fill slower to reduce the effect of large pressure swings when flushing the toilet (no more screaming from the shower after every flush). If this is your problem, then it's "working as designed."
When operating any cutoff valve, there's always the possibility that it will fail, particularly with the old gate valves. Many have been moving to the quarter turn ball valves for the shutoffs within the home due to their increased reliability.
The most likely cause of a slow fill is not completely purging the fill valve during installation. After all the internals have been connected, there is usually a step to open the top of the fill valve with the water turned off. On one of the more common fill valve styles, this requires the float to the lifted up while pressing down on the top of the fill valve and then give it a quarter turn like you would a medicine bottle. With the cap off, you clean and set aside any small parts, put a cup over top of the fill valve, and turn on the water supply for 15 seconds. The cup is to keep the geyser from shooting all over the bathroom and stay inside the toilet. You then reassemble the valve with the above procedure in reverse.
For the overfill pipe, the only position inside the toilet that matters for this is the height. It needs to be below any openings inside the toilet tank (e.g. a gap in the back or an opening for the flush handle), and then you adjust the float to stop below the overflow. I tend to give myself between 1/2" to a 1" margin of safety for each of these heights. Then the refill hose goes into the overflow tube. The fill valve sends water two directions, most out of it's base into the tank, and the rest out the refill tube to refill the toilet bowl itself. If you remove this refill hose, the bowl will not refill and you'd get sewer gasses coming out the toilet when the water level drops the slightest bit.