Airport Extreme lan ports

No changes are needed.

Switches are cheap consumer item now.. And you can buy the cheapest or the nicest looking or ones with the ports on one side and the lights on the other. Many many variations. Basically they are all the same internals. I like ones with metal cases. They tend to work better from heat - cooling point of view.

They mostly come in 5 8 16 24 48 ports.

Buy 8 port gigabit as it is just right .. 5 is too few since you lose 2 ports plugging in a switch. And for most people 16 is too many. It is better to plug everything into the switch.. and just use a single ethernet from switch to Airport LAN port. The reason is backplane speed of the switch is faster. Although for most home usage it does not make any difference.

Yes, you can directly connect an Ethernet switch to any one of the AirPort Extreme's LAN ports to effectively increase the number of ports. The Extreme has an unmanaged Ethernet switch built-in. You would just need another 1 Gbps unmanaged switch for this.

30M is about the limit for gigabit on early Cat5E. It will still work but can be slower at 100Mbit.

Cat6 or Cat7 are rated to full gigabit to 100M. And usually Cat5 will also work fine even to gigabit if it is terminated properly.

A lot depends on the cables. The standard is 90M of solid core.. that is less flexible cable in the walls. And 2x5M stranded patch cables at either end. People try and use solid core for patch cables or stranded over long lengths and then get into trouble. Professionally installed cabling will give you very few issues in regular house dimensions. Even DIY is not that hard.

The AirPort Extreme is compatible with CAT 7, but using it will not improve the performance of the AirPort Extreme.

Do I need to connect the New Switch to the Modem first ?

The switch must connect to the AirPort Extreme......not to the modem.