I move mine to pots as soon as they start to sprout at all. The problem with leaving them in trays like that is, as roots develop they will tangle up with each other and you will have to tear them apart.

Yes you can plant them deeper in the pots - you will certainly plant them deeper when they go outside. Cut off any leaves that have unfurled if they are going to be below the soil surface.

Its exciting to see them waking up isn't it?

Since you mentioned that you are a newer gardener I'll point out something that you may or may not have fully understood. If these were tubers that you acquired from a reputable source (mail order, store, an experienced gardener, etc.) you might as well just pot them up individually to start with. The reason some people (myself among them) plant lots of tubers close together in shallow trays and then later move them along to pots is to test them for eyes. When dividing your own clumps it is sometimes difficult to tell whether a division will have an active eye or not. I divided one clump that had over 40 tubers but I didn't expect to be able to carve them all away from the main stem with eye tissue. I just did my best to get a piece of the stem where an eye was likely to appear and then I planted them all in a shallow tray to test for eyes before potting up the active tubers.