your question is tied to the scalability issue, at the moment we don't have these problems but in the future we will.
1- a big amount of transaction doesn't mean necessarily huge calculation and processing because the mining process doesn't depend on the transaction quantity but in the network difficulty.However as you said this growth will impact the network latency. But the system is designed currently for a specific throughput to increase it we need to find out an efficient scalability solution, like segwit for the Bitcoin.
2-currently the size increases 1GB-2GB per month for about 4 tps. if the transaction volume increases (currently we are at 40% of block capacity utilization) the situation will change and we will face a storage problem (maybe in a far future). the storage technology evolve in time and we may see the cost's drop of the helium disks or even biometric storage(for a far future). Besides, if the cost of storing the blockchain becomes so expansive this may lead to a centralization problem,as vitalik points out:
excessive size leads to more centralization which leads to more fees
due to monopoly pricing
but there is some solutions trying to deal with the scalability issue like the sharding