On sets and reps:
Hitting the point of exhaustion on your last rep is good. However, not all "exhaustions" are equal.
Let's say you lift 3 sets of 10 reps one session and 5 sets of 6 the next session. Both days you lift the same amount of weight with rest periods set so that you hit the point of exhaustion on your last rep of your last set.
Yes, both times you're exhausted. Yes, either way you're doing the same number of reps (30). However, the effect will be quite different. Hitting exhaustion on the 5th set means that your first set probably wasn't productive and your second set was less than ideal.
When people talk about "periodization" where they change the elements of a workout they usually change the weight (as a percentage of your one-rep max), the number of reps in each set, and the amount of rest between sets. However, the number of sets usually doesn't change and with good reason.
3 sets is the rule of thumb because it's generally the most productive for most people. Heck, many trainers advocate one-set systems, though this is usually only because it tends to increase compliance (less boring for the client) without sacrificing too much of the gains. I don't know anyone that recommends more than 3 sets for anything other than attempting to break through a plateau (and even then, usually as a last resort).
For your goals I'd recommend 3 sets of 12 reps with a rest period of 4 minutes or so for your large, compound lifts (squat, deadlift, bench) 2 minutes for any simpler exercises you're doing (flys, curls, etc). Weight determined by what you can complete with good form.
It's not as good as having a structured, periodized workout, but it will get you better results than doing 5 sets.
I strongly disagree with the suggestion to cut out the heavy cardio. It's easier, sure, but it is less effective. Walking for 30 minutes is a great first step to getting on the right track, but with your goals and access to the equipment there's no reason not to take advantage of it.
Most people, when they go for a brisk walk, won't get their heart rate over 90 BPM. Depending on your age your target heart rate is probably around 145 BPM during cardio, held for 20-30 minutes. Now, keep in mind the following.
Calories burned during aerobic exercise are a strict function of oxygen consumption. Oxygen consumption is in turn related to heart rate, but not linearly. As your heart rate goes up your stroke volume (amount of blood pumped per heart beat) also goes up. As a result, working out at a higher heart rate means much higher oxygen consumption and way more calories burned.
If you want to throw away close to half of your calorie consumption (and a big part of your heart health) to trade the elliptical for daily walks, I can't stop you, but with your goals it doesn't make sense.
On the lapse:
Don't sweat it. Lapses happen, and the odds were definitely stacked against you with everything that was going on with your family and job. That being said, take a lesson from this.
Try to look at everything that went wrong, everything that made it harder for you to work out. Use this to develop plans for the future. Could you have possibly gotten your cardio in before heading over, or while you were there if you packed an extra bag? What about before or after the job? Whatever answers you come up with, remember them and plan ahead next time.
On the elbow pain:
Good job not aggravating your elbow. Seriously, I'm proud of you. A lot of people get in the mindset that they need to push through the pain and end up causing injuries that, while not permanent, are serious enough to knock them out of their routine long term.
Listening to your body when something hurts or even just feels "off" is incredibly important in general, but ESPECIALLY as you're adapting to a new workload. There's a good chance it's nothing more than some tendon inflammation, but pay attention to it. If it doesn't improve or get worse, go to a doctor. Worst (Best?) case scenario you're out your ~$10 co-pay and get told to RICE it. Best (Worst?) case scenario you catch something like a minor stress fracture, bone spur, or subluxation/dislocation early and prevent it from becoming a serious injury.
Best of luck to you. Hang in there.