Is the price of BTC transactions driven down by the influx of miners? I'm wondering why Bitcoin transaction speed doesn't increase (and cost decrease) due to more people becoming miners all the time. Is it because the number of transactions is increasing even faster?
No, it is because those are unrelated values. The number of miners (more preciselly the cumulative hash power) influences the security of the network, but not the speed of mining the blocks (in long term). In short term (within the current window of 2016 blocks), it can influence the block generation speed, but after the window completes, difficulty retarget happens, so that the block generation speed goes back to designed avg. time of 10 minutes.

The transactions can not be confirmed faster than blocks are generated, thus block generating speed is very closely related with transaction [confirmation] speed. 

Transaction speed is also influenced by number of transactions waiting to be confirmed, but that again is unrelated to number of miners.

Similarly the price is more or less unrelated (assuming enough miners to avoid monopoly). The price (per byte of serialized transaction) is mostly affected by the demand - i.e. the number of other transactions waiting for confirmation and their size and fee used and some other subtle properties.