Can I open a new channel, left it open and use it multiple times to pay other people?

Yes

I know that if there’s any kind of connection (through channels) between the person I opened the channel with and the new person I can pay the new receiver recycling the old channel.

Yes

But there’s any way to use the same channel to pay multiple people or I just have to open a new channel for each payment to different person and pay the fees?

No, you can use the same channel multiple times for everybody on the Lightning Network whenever you want

I know I can avoid this with non-custodial wallet but I would like to do this while owning my own Bitcoin

Yes, you can do this with your own node if you want

Also, can I avoid the fees by running my own node?

The fees on the Lightning Network are very low. And there are many people having zero-fee channels, you can specifically open your channels to those people if you want