Upvote:4
If you are not checking bags, whether or not you will need to re-clear TSA security on a domestic connection at a US airport depends solely on the configuration of the airport itself.
In the case of LAX, you will not need to exit the secure area, as there is an underground tunnel that connects T5 and T6, as summarized on the Inter-Terminal Connections page at the official website. A long time ago, the tunnels were the primary way to get to the gates, but that's a matter for history.
In T5, the escalators and stairs to the tunnel are located between gates 53B and 55A, near the food court and restrooms. Go down to the tunnel level and follow the signs for T6. It's not a pleasant walk, but it should only take maybe 5β8 minutes or so. In Terminal 6, the stairs/escalator bring you up between gates 64A/65A and 64B/65B.
To summarize the overall situation at LAX as of November 2018:
As to your second question, if you book separate tickets, you incur all risks involved with the connection. If your first flight is delayed or canceled, you have no rights with the second airline, and will only be able to plead for mercy as a customer service gesture; quite often, you will only be able to get rebooked on a space available basis (i.e. standby) and have to hope that there is space on a later flight that day.