Just my opinions here...
Basic blue book value on the 2001 Rodeo is roughly $2500 to $3500. So the repairs are more than that? If the repairs are made, would the car in basically good enough shape to last another couple of years?
If that is the case, fix the Rodeo. Basic math wins on this one. $4000 is less than $8-10000. You have to come at the idea that while the basic value of the car isn't "worth" the cost of the repair, you're proposing to spend even more money than that. If the car is otherwise in good shape and will run for many more years with basic oil changes and little fixes, it's actually worth paying out for the repair than saddling yourself with a more expensive outlay. Plus, cheaper insurance and you're familiar with the car's quirks and probably don't care about a few dings and the like.
If the car is to the point of being very unreliable or this breakdown is just one of many, the a new (used) car probably would be a good idea.
There really are only 2 good arguments for financing: either you get zero interest, or you get cash off the top for doing so. If the second is the case, then do it, but pay the car off as soon as you can. READ the contract carefully and ask however many questions as you need.
I just got a new car after my 17 year old vehicle finally had so many breakdowns that it left me stranded at the side of the road several times in a row. When I purchased, they offered a $500 incentive to finance. I read through the contract very carefully to see if an early payoff was penalized, then did the financing and then went home and called the finance company to re-confirm there was no penalty (was verbally told there was, but it wasn't in the written contract, and the finance company confirmed that "that's what the sales guys say, but there is no penalty) and then paid off the car the same week - meaning I snagged an additional $500 off for just doing the paperwork. Otherwise, I would have paid cash.
Buying a house in the summer months (I was told) isn't usually the best time, since that's when "everyone" tends to be out looking as the kids are out of school and moving them from one school district to another isn't as painful for them... so the market is supposed to be hotter during the summer months (pun intended?). If you wait until the fall/winter months, it might mean that the houses on the market have been there for a while... and more eager to deal. You can usually search the realtor sites to see how long a house has been on the market and also if they've lowered the price (good indicators that they'd be willing to negotiate more). Also, do the research on first time homebuyer incentives and grants (google "first time homebuyer and maybe your zip code?). There should be many out there.
Also, check in with banks to see what their mortgage rates are, and if you're seriously looking, maybe get preapproved (not qualified) so you know what range of home you can afford. Use that info to look for a house that costs LESS than the amount they approved you for, tho.
You said you wanted to buy if your contract (hopefully) gets renewed... that's a concern as it sounds like you'd be out of a job if it doesn't get renewed. If that's the case, you should definitely wait to purchase a house. You do not want to take on that much if you're unsure of your income.
So basically if it was me: I'd either repair (first choice) or replace the vehicle (if it's just falling apart), paying cash unless the financing is a great deal, and hold off on the house purchase for a while until you're sure you've got a steady income and the home sales market might be more in your favor.