PA28 pilot experiences a gear malfunction after takeoff that is successfully resolved in the pattern prior to landing. While thinking about the gear problem during taxi to parking a runway incursion occurs resulting in a go around.

Narrative

I was flying two EAA Young Eagles and their Grandfather. Their dad was on the ground taking pictures. On short final to [Runway] 15L (I was cleared to land); I heard Tower tell a twin to line up and wait on [Runway] 25. The twin read back 'cleared for takeoff 25.' After a very short pause; I said; 'Is that line up and wait or cleared for takeoff.' Immediately Tower said in a loud; firm voice; 'Twin line up and wait.' The twin read back correctly and I landed with no problem. I think this was the beginning of the almost accident chain because it rattled me a bit. Upon landing; my wife told me the dad had not gotten the pictures he wanted and could I go around the pattern one more time. We took off on 15L with a left pattern. Upon clearing the runway; I pulled the gear up. The 'in transit/gear unsafe' light came on; I turned left crosswind. The light stayed on. I turned downwind. The light stayed on. (I did not notice or at least cannot remember the status of the gear lights.) I put the gear handle back down. I think the gear unsafe light went out at that point; but am not sure. I was getting more rattled because I got two greens; but the nose gear light was NOT on. I pushed the red emergency down handle; still no nose gear. I announced to Tower and prepared to fly a race track over the airport. I pressed the emergency handle a couple more times and tapped the nose gear light. Suddenly; I had three greens. I made a good landing and all was well.I thought I was fine; but apparently I was rattled still. 'Taxi Bravo; hold short 25.' I read it back and started down Bravo. I've done this a 'million' times over the past 6 years. It is always 'hold short 25.' But I was thinking about the gear issue and blew right past the hold short line. I did look to the right; saw the jet on short final; and stomped on the brakes just as ground staid 'STOP.' I didn't enter the runway; but was clearly and completely past the hold short line. The jet had to go around. I got clearance to continue and the flight ended well. They always say 'fly the airplane.' That applied to TAXI the airplane just as well. Instead of paying attention to taxing and holding short; I was replaying the gear issue in my head trying to remember exactly what happened and what lights were on at each event and exactly what I was going to tell the A&P. Plenty of time for that later. Luckily a sharp controller or two and a sharp Jet pilot and the fact that at least I looked before crossing prevented what could have been a much worse ending.