There is no need to mess with the cap number. If 1 BTC were to have a value of let's say $10 million (which, I'm sure you'll agree, most people would consider an impossibly high value for it to reach), then grocery purchases, if we're to assume will have costs of between $1 and $1,000, would be happening within the ten millionth to the hundred thousandth range (0.0000001 ~ 0.0001) in BTC terms.

In that case, instead of measuring in terms of whole BTC units, which would be really confusing and counter-intuitive, we'd likely be measuring in ten millionths of 1 BTC unit (which is 10 satoshi, btw) so that the bitcoin unit costs closely resemble the dollar unit costs. That way, a $100 grocery bill will equal around 100 bitcoin units.

Depending on whether bitcoin is to be the world's future solution to digital cash, it may not have to increase transaction speeds that much, if at all. If it's merely going to be a digital gold (store of value), then transaction speed isn't much of an issue. It will definitely need to become much faster if it will replace physical cash.

Bitcoin is going to increase block size to 2 MB this November, which will increase transaction speed from around 7 per second to 14. The downside to this approach is that the memory space required to hold the entire history of the bitcoin blockchain (requirements for a bitcoin node) will also double in rate (with the doubling of block size), so there is a limit to how big the blocks can be in terms of Mbs.

I don't know exactly how it works, but Lighting Network, which is also going to be implemented this November, supposedly offers a work-around solution to the block size limitations, through the use of "off chain" transactions (that, I assume, combine many transactions together outside of the network and send them in large batches to the bitcoin blockchain -- if you're reading this and find an error, please feel free to correct it). This, as far as I'm informed, makes it possible for bitcoin to handle many thousands and even millions of transactions per second in the future.

As for the cost of transactions and the mining cap, I direct you to the following quote from this source:

Transaction fees are voluntary on the part of the person making the bitcoin transaction, as the person attempting to make a transaction can include any fee or none at all in the transaction. On the other hand, nobody mining new bitcoins necessarily needs to accept the transactions and include them in the new block being created. The transaction fee is therefore an incentive on the part of the bitcoin user to make sure that a particular transaction will get included into a block.

It is envisioned that over time the cumulative effect of collecting transaction fees will allow somebody creating new blocks to "earn" more bitcoins than will be mined from new bitcoins created by the new block itself. This is also an incentive to keep trying to create new blocks as the creation of new bitcoins from the mining activity goes towards zero in the future.

Because of deep technical reasons, bitcoin block space is a scarce commodity, getting a transaction mined can be seen as purchasing a portion of it. The price of block space is set by supply and demand, although in the real world the supply of space for transactions is extremely noisy, because more becomes available (and has to be immediately consumed or it’s lost forever) every time a block is mined, and block mined is an intentionally random process, that randomness being essential for bitcoin's operation. Demand is random and cyclical. Random because each transaction is generated individually so the total amount is noisy (although that averages out to be somewhat smooth at scale) and has both daily and weekly cycles, with more transactions done during the day than at night. Demand can also be affected by speculative movements in the exchange rate.

That last paragraph says to me that each block needs to be sufficiently large so that the block space isn't so scarce and the demand can be more easily met by the miners.

Hopefully that covers everything that you were trying for.