Can a patch panel support both cat 5e and 6a?

Have to be cat 6, that is backward compatible with cat 5.

Yes, it's fine. The actual copper strands are the same size, it's such as the twist and pair arrangements in the cable structure that are different.

I'd highly recommend using metal body "keystone" socket inserts for patch panels.

You can get them online relatively cheaply and the termination is a lot better than most other types of socket.

You attach the socket to the cable end then just snap it in to the back of the patch panel strip or a wallbox faceplate.

There was some place on ebay selling big batches of ones that look like this a while ago - they are excellent, I've not had any problems after using a couple of hundred.

https://www.ebay.co.uk/itm/Luggar-Cat-6-Toolless-K...

They do not need a punch down too either, you just arrange the wires in grooves in the plastic insert then push that on the rear of the body, trim the excess ends and close the clamshell end to clamp everything solid.

A cable tie forms a strain relief.

Or these, ten plus some wallplates at a reasonable price:

https://www.ebay.co.uk/itm/5x-Cat6A-Netzwerkdose-D...

We also use a plastic version that looks something like this, for wallbox sockets where there is not enough depth for the cable to come out the back of the insert:

https://www.ebay.co.uk/itm/2Pc-UTP-CAT6A-Network-M...

Once you start using keystone fitting stuff you will never switch back..

The patch panels cost next to nothing compared to conventional ones, as they are just bits of metal.

Better ones have a rail spaced back a bit that you can use to tie down the cable clear of the socket itself for additional strain relief.

See below.

We mainly use Netgear managed / stackable switches, which fit in moderate size rack cabinets with the patch panels etc.

See the third link for the typical cabinet style, though we normally use 10u ones to allow plenty of space & for cable management strips.

Unless you are going to have dozens of servers, I'd keep them in conventional, large, desktop cases. It's a lot cheaper as you do not need special parts and the airflow is far better, and with easier access for maintenance if needed.

The only practical reason to use rack-mount cases is if the machines physically cannot fit otherwise. The costs are higher, they are noisier and reliability is lower, long-term.

Most of the newer patch panels that I've checked out mention that they're backward compatible and list the older cables.  So, I'd look for that wording on the gear you want to use.