I think it is a Lyon and Healy model as mentioned above although if it is the Lange Princess it is worth a bit more I would say. I used to play a Lyon and Healy short scale tenor until recently. (I have forsaken the banjo now to concentrate on the full mandolin family and guitar, and because I am a gentleman!)
Stew-Mac have a variety of replacement tuners on their site. The plantetary type are more expensive but do not slip. On my Lyon and Healy (and an earlier Vega N style) I used good quality friction-type tuners which are still available and which worked fine and, to me, seemed to be more in keeping with the provenance of the banjos.
The "normal" tuning for a tenor banjo would be CGDA but most Irish players use GDAE an octave below the mandolin, also as explained above. I am used to both playing the mandolin, alto mandola, octave mandolin and mandocello but GDAE may be more useful to you playing with other Irish musicians.
String gauges are a matter for some personal preference and to suit the individual instrument and scale length. I think you will need to experiment until you find what suits you best. I have used from .010P, .014P, .024W, .034W to .012, .020W, .028W, .038W (similar to Eric the Red above) according to the instrument... generally nickel wound (which last very well).
Still looking forward to meeting up with you sometime in the not too distant future and having a blast (and a few pints)!