Adding and improving upon what Houlio answered:
- Does the character automatically get an LR? No. You need to purchase it using accumulated vis or through seasons of service. The cost is usually 1 pawn per magnitude when asking a specialist to do it, plus the actual vis needed to perform the ritual. Members of House Mercere often benefit from a slightly reduced cost.
- Any special rules for finding/binding a familiar? It takes at least a season to find a familiar. Treat the season when it is actually found as a low-experience story (5 xp or less during that season). Grog-level familiars (usually under your control) have a base Might of 9 or less (before Size adjustments and Qualities/Inferiorities). Companion-level familiars can be more powerful but must be played by someone else (not always the same player and does not count as their companion slot).
- Do I assume the stats of his new covenant for post gauntlet development or is he still a Durenmar magus? No matter where he is, consider he has at best access to the equivalent of a basic lab in a +3 aura. Your narrative might justify a lower aura or poorer lab, if you want.
- is the character a master at the end of those extra years? Automatically? Under certain conditions? Never? No. Becoming a master is a significant milestone. Receiving any of the tokens should happen in play, although you can have performed some of the groundwork during pre-play advancement. The rank only has significance in the Rhine, so lack of it does not bar your magus from other achievements (such as taking an apprentice) if you craft your narrative around that.
- Does he join the new covenant right after gauntlet, or at the end of those years (this, unlike the question before it, is a plot question, not a mechanics question) No. He joins Tugurium at the end of those years. The narrative should certainly integrate elements explaining why he would be interested in doing so, and could also include some sort of link. Clusius' narrative (starting here) contains many possible elements that can be used.