I would ask the mechanic why does he suspect the right tie rod - Did he verify that the joint completely worn out? Did he check the alignment to verify that it was out of spec?
How is the tire worn - worn unevenly? cupped? feather-edged? etc. Anything else seems out of the ordinary, harder to steer, pulling at highway speeds, etc?
$250 to change a tie rod, rebalance the tire up front? didn't mention anything of an alignment - that sounds a bit off. If the tie rod is indeed bad and the tire wear is excessive, you should really replace the tire (possibly both tires - keep them as an axle set) before you do an alignment and wheel balancing.
As for the mileage - 35MPG on highway is pretty decent. 20MPG in the city is a little lower than average, but not out of the ordinary, depends on the driving conditions. Possible that the excessive tire wear and possible alignment issue is sapping some mileage in the city, though getting 35MPG on the highway makes it seem like you don't have a serious alignment/tire issue yet.
Not sure what the mechanic meant by jerking vehicle, you mean the engine rocks on its mounts when you shift in and out of gear? If the whole vehicle shudders when you shift in and out of gear, could be worn mounts or it could be the engine needing a tune up. $200 for parts and labor is about ballpark for this type of work - but you have to verify that mount is indeed bad. At this mileage, it is very possible the mount is worn out. You can check yourself by brake torque the engine briefly and watching how much the engine moves forward and backwards. It will move a bit under normal conditions, but if it moves enough to bang into the hood or seemingly rotate more than a couple of inches - could be bad mounts. Usually worn mounts are coupled with excessive steering wheel and chassis vibrations.
Lifespan of the car - hard to say, with any certainty. Maintenance is a big part of that equation - letting even small things go can end up costing you a considerable amount down the road. If it worthwhile to put in a couple of hundred of dollars into this car or start looking for another one, that is completely up to you. I've seen posts of owners that have less than 20K-30K miles on the car before something catastrophic happened to the transaxle or gearbox - and others that have gone well over 300K miles with no major mechanical issues. As long as you stay on top of the maintenance, what I've found is that is much more likely that the car will be disabled/destroyed in an accident before the powertrain fails on these cars. My own Matrix XRS was just touching 120K miles and running perfectly, before it was rearended and declared a total loss by the insurance company. That car could have easily hit 200K-300K miles, in my opinion, actually ran better at that point than when I first purchased it used 5 years ago. Current replacement 2009 Matrix is at a little more than 25K miles, and trouble free - aside from tire issue, and my current 2002 Corolla is over 185K miles and running strong - 200K-250K no problem.