My $.02: If your 5k time is above half an hour, you aren't running. You need to focus on actually running. If you're trying to jog and you slow down beyond a 10 minute mile pace, try just walking for a bit until you have enough breath/energy/moxie/whatever to get back to a quicker pace. There's no shame in walking, and jogging extra slowly just wears on your joints without giving you much of a benefit.
I'm no expert on weight training, but as far as non-interference goes, I really think that in your situation a personal trainer would tell you to go ahead with the cardio training even if it means you have to scale your strength training back somewhat for a few months. At this point getting in better cardio & jogging shape will probably help your weight training more in the long run than all-out weight training will. If you're not comfortable with that thought, find a personal trainer and discuss things with them.
Two kinds of runs to try: 1) Get so you know what a <=9:00 mile pace feels like (even if right now you can't keep it up for a mile) and go as far as you can at that kind of pace. Don't let yourself slow down; pour more energy in to maintain the pace until you really can't do that any more, and then stop and walk it off. Work up the distance over time. 2) Intervals for working directly on speed. You may dislike them, but they're effective. "Equivalent performance" target paces for different distances can be calculated at http://www.mcmillanrunning.com/calculator
Do one or the other of these at least three times a week. Be sure to give yourself a rest day every week.
For endurance work on the other days, rather than slow jogging at a >10 minute mile pace, keep up the hiking and throw in some cycling. (If you're on pavement, thin, high-pressure tires can make biking a lot more fun.) Swimming lots of laps can be quite good too, but most folks don't particularly enjoy that, and if you don't find a way to enjoy your endurance work it's hard to be consistent about it.
Since exercising much less than half an hour is not really good endurance work and running >10 minute miles is kinda pointless, meeting your goal is basically the point at which using runs as endurance workouts really starts to become viable.
Your goal is extremely doable if you're consistent for a while. Good luck.
I've been in a somewhat similar position- at one point I was quite sick for a couple years, and after I recovered I could barely do a 9:00 mile. It took me a good while to work up to the point where I could keep a sub-nine pace for 3mi, largely because I wasn't as consistent as I should have been. Not too long ago, I did a half-marathon with sub-8:00 mile splits, and one of my goals now is to do a 12:00 2-mile (plenty of work before I get there tho).
PS. GuruBill talked about improving form and avoiding injuries by avoiding heel-striking. Just shooting for a 5k you can get away with heel-striking, and making the adjustment to a midfoot/forefoot strike would take some time. But making the switch really is worth it. Though he says you don't need to have a special shoe to do this, the truth is that it's very hard to train yourself to avoid heel striking when your shoe has a lot more padding in the heel than in the front.
The difference between the thickness at the heel and the thickness at the forefoot is called the drop of the shoe, and having a high drop is not only bad for your form but also for your posture (just like high heels). Running shoes used to all have moderately low drop, but in the 80s and 90s they started to pad the heels more and more. The extra few mm of padding can't really take the impact of hitting the pavement off your joints, it can only obscure the pain which is your body's signal trying to tell you "Hey, don't run that way!" and thus actually makes injuries more
So it can be to your advantage to get a low-drop or zero-drop shoe. There's plenty of either these days, but the first examples that spring to mind are the Brooks PureConnect (low-drop) and the Altra Instinct (zero-drop).