As @Bitcoinnation points out, the wallet issue with size you are focusing on is if you are running a full Bitcoin Core none, being the size hazard easily overcome by adding a simple external USB disk as a repository (I use a 3Tb external USB 3.0 disk and it works smoothly). You obviously do not need to run a full node to use a BTC wallet.

As for the costs of mining, it depends on the crypto type itself and the underlying algorithm and difficulty established. There are well known cryptos such as Ripple, Cardano, Stellar, Eos and Neo that are premined, and therefore have no mining associated cost.