Do you already have a motorcycle or would you be buying one? Depending on your climate, is a motorcycle practical 365 days per year? Do you do other things you'll find yourself wanting a car for?
10MPG up from a Civic probably means going from 35MPG to 45MPG (or similar), and that's not saving you much gas unless you do a ton of miles. It'd be a nice bonus if it otherwise pencils out, but I wouldn't have that be a reason.
If the (non-dollars) cost of the car breaking down is pretty low (you're not that far from work, or you have public transportation as an option), I'd probably just hold onto it and keep driving (and keep up on all maintenance!), but have a plan for what to do when it does give out.
I would be taking the money from the sale of the Civic to purchase the motorcycle for cash. As to weather, we seldom get snow/ice, rain is the worst offender and when I had a bike before, I used the car on those days. I would need to pickup a pair of rain pants to round out a complete set of gear to ride in any temp down to the 20's. I rode all winter a couple of years ago on the old motorcycle I had.
We have a 2007 Hyundai Entourage that is payed for that I can use for any hauling or the like. My wife drives that as transport for her and our 3 kids. She works at the school they attend so they all ride in that 5 days a week to/from school.
Mpg would go from 30 to 40-ish, and work is ~8k miles a year of round trips. Work is ~17 miles from home. The housing market near the office is ~2-3x what I pay here. No public transportation is available unfortunately.
The text book answer is: ASAP/now.
But, the math answer is: it depends how much available credit you have on the credit cards.
If you've got $1,500 (or so) in available credit, that can be the bulk of your 'emergency fund' (I call it 'life fund' because emergencies may happen but life does happen). You still want/need $200-$300 in savings - which also should prevent any monthly account fees. Some people need this in a separate account.
Note that this is only viable if you can/will only use the credit for actual emergencies. I've heard that some people put the cards in a bag or container of water and put it in the freezer.
I have ~100k free on cards. I have several 15 year old accounts etc that I leave open and randomly use for 0% transfers if the terms are agreeable.
I think we have ~500 in actual savings. I have not done the money for 15 years as I was a huge spender and my wife was/is not so I let her take the reigns. Reading the blog really got me thinking about my bad habits and I am serious about turning this ship around. ER is nice but my major focus is getting rid of the "optional" debt(CC,LoC,SL) and then looking towards structuring retirement.