My kit includes ICs labelled :
• M74HC595B1
• M74HC165B1
The labels on the PCB are :
• 74HC595N
• 74LS165N
Do I have the right chips ?
Yep - The important part is the xxxx595 & 165 in this case - particularly if they came with a kit, if you selfsourced you need to check the data sheet if they can handle more than 5v … Anyhow, all parts that come with the kit are vetted & checked, no need to check them all by yourself 
Yes.