Can I add more yeast during primary fermentation?

Yeast is good up to 110F. Death STARTS at 120F and Total Kill is at 140F. I've had to add yeast daily before due to buying dead yeast. Best to toss your yeast in a glass with 3 TBS of sugar and half fill with H2O that's no hotter than 105F. As a matter of a fact use 100F as a SAFE range everywhere. GENTLY stir the mix to break up clumps. LEAVE IT ALONE 15 Min.... If not a significant reaction... do over till you find a good start. Add to your fermenter and away you go.

Have been doing cider for sometime and decided to do a BIG double IPA. Threw my yeast in at 120 f after watching a YouTube video that did not even show the cool down temp process...low and behold I had a pillow near the carboy the entire first night and the batch was, well...not to excited with me. Asked my local home brew supplier and he advised to add another 2 packs of yeast to my carboy that I most likely burnt initial yeast. So farrr.... 3 days of glorious action with wort foam and 11th day percolation rate of a bubble every 90 seconds!!

Healthy yeast, pay no attention to the mass burning of your species as they float by you...thus the circle of life

Agreed with @brewchez.

The SG of 1.014 tells you that a lot of fermenting has already happened (though it is odd that you saw no airlock activity; what about yeast foam on the surface or stuck to the sides of the fermenter just above the liquid level?). With an O.G. of 1.040 you'd usually expect a bit more attenuation, down to 1.010 or 1.012, unless you've made a relatively unfermentable wort (lots of crystal, or high mash temp.)

Sometimes a highly flocculent yeast can be swirled back into suspension and this can help it finish off the fermentation a bit quicker.

It is possible that your fermentation happened so quickly (you say you pitched at midnight) that it was over and done with by morning time.

Probably your two biggest risks from high temperatures like this are the development of off-flavours, or having an infection set it. If your sanitation is really good you can perhaps get away with the latter. The former is very much a function of the yeast strain and I've never used S-05 so I have no idea how it behaves at high temperature (I did once find out that S-189, a lager yeast and my favourite one in fact, gave a good clean fermentation at 23C - wow!)

If you really do want to raise the alcohol strength, I would add additional sterile wort and mix or agitate it well. Even if you brew all-grain the easiest way would be to boil up some water and malt extract. Adding say 5 litres of 1.080 wort to your 20 litres of what-started-out-as-1.040-wort, would give you 25 litres of wort that would have an equivalent OG of 1.048. (though my thoughts on this are, if you are in the middle of the summer in Australia perhaps a 1.040 beer would be nicer than a stronger one)

You can add more yeast anytime if you like, but 1.040 to 1.014 sounds like its done fermenting to me. The beer isn't going to get much more fermented than what it is now.

The beer would have to be pretty hot for the yeast to get completely killed off.

There should be plenty of yeast left to carbonate the beer. If you really feel that yeast is the reason why your last batch didn't carbonate, then a half a pack of dried yeast rehydrated in a cup of boiled and cooled water can be added at bottling right in the bottling bucket along with the sugar solution.

Adding yeast to a cold carboy of beer is just going to force the yeast into a dormant state and shock them. So I don't understand why you would consider that. (Maybe its not clear from the ?)

Need some more info. What type of yeast and what recipe is this? What temp did you ferment at? It sounds like you may be close to a FG for some styles. And may still have a leak in your system if you are getting no airlock activity. I wouldn't recommend re-pitching with that low of a final gravity, if you are just looking for a higher alcohol content you would have needed to start with a higher OG to begin with.

Short answer: Yes you can. You'll need to make a starter and get that yeast actively fermenting before you pour it into your main fermentor, though. That's now a fairly hostile environment, so adding them at the peak will help them deal with the low sugar & high alcohol environment that you'll be putting them in. I'd suggest a fairly neutral flavored yeast (if you're making an ale, US-05, WLP001, WY1056)