Beth 30,
I ran for several years in Kuwait year round with a number of colleagues. The trick for me was to be wary of sun and humidity, and be careful about hydration.
Running in dry conditions up to 45 C was fine at night for up to say 40 - 45 min (10k) but in that time we would drink up to a 1 litre of water whilst running and the same again afterwards. Above 45 C things got difficult, but luckily it only goes over 50 C for a few days a year and then only during the day.
Humidity is the real killer, and I know this is a problem in Dubai. 35 C and high humidity (80%+) is the same apparent temperature as 45C and low humidty (20%). From charts I have seen and personal experience, these are certainly limits above which care is needed to avoid getting into difficulty.
Avoiding radiant heat from the sun I always find to be a great help. There is a huge difference between running at 5pm in low sunlight (or at night) and running at 1pm in the full glare.
Hydration is vital. Lose more than 3% of your body weight and you get into trouble ... from experience this is no fun at all. To avoid this, drink masses and don't push it until you are comfortable with the conditions. The body does train into heat - after a couple of weeks you sweat sooner and at up to twice the rate from articles I have seen. But you still have to keep drinking ! 0.25 litre every 15 min is what I normally drink in hot conditions.
Longer runs at this time of year in the Gulf are also fine. 30 - 35 C in the evening can be a great time for longer outings of up to 20 km or so (humidity permitting !). For these, I personally find in addition to water, a sports drink is a very good idea half way round and at the end.
Good point about heart rate, I always found a heart rate monitor to be a helpful early warning indicator of stress.
At the end of the day it is a personal thing. By listening to your body, keeping hydrated and gradually building up endurance in the conditions you will be surprised at what is possible.
Good luck
M