Electrum is the only bitcoin wallet so far that has smaller fees compared to other wallets.
As much as I think Electrum is great, its fee is no smaller than any other wallet which lets you pick your own fees. The wallet you use does not make a different to the fee you pay, unless you are using a wallet which sets the fee on your behalf and doesn't let you change it (in which case you should change wallet).
I thought downloading through the website is already a safe means. How does one verify this again? Thanks.
Not necessarily. The website or their file host could have been compromised and you could therefore download a fake version, even from the legitimate site. It's unlikely, but possible. The only way to be completely sure you have a non-malicious version is to verify the file. The post you quoted contains a guide on how to do this.
A low bitcoin fee means you will wait longer for the transaction to be confirmed.
Not necessarily. If the mempool is completely emptying with each block (as it frequently does many times a day at the moment), then a fee of 1 sat/vbyte will confirm just as quickly as a higher fee. In these scenarios all you achieve by paying a higher fee is losing a little bit more money.