The P-59 Airacomet was probably a pretty safe a/c to transition in (with large, thick -albeit laminar flow- wings--resulting in good engine-out and stall characteristics, freindly handeling, and modest top speed for a jet), this was the only role it ever served operationally, and it was the same characteristics which made it unsuitable as a combat a/c. The Yak-15 would be similar. (although it had decent performance over piston a/c, although very poor range)
Another thing to note is that the He 162 wasn't expressly designed for the "people's fighter" competition, being largely based on a light-weight fighter design privately developed by Heinkel. So the design had been around much longer than the begining of the RLM's request. In fact Heinkel had been trying to get a fighter into service for much of the war but the RLM didn't particularly like Heinkel (somewhat frustrated with private projects like his jet program instead of focusing on Bomber production and devlopment, as well as for political reasons- having a strong bias tward Messersmitt pluss Heinkel's personal dislike of the Nazi regime) and they only realy chose his project as it was farther ahead in design and higher performing than the compeditors.
Also, as 100 Me-262's would certainly be more useful than 100 He 162's, rember that the He-162 was about twice as easy to produce and uded only one engine so you could easily have twice the a/c. (it was also faster, more agile, ans a better climber) Although 2x MG-151/20's are weak for an interceptor 2x MK-108 (although with only 100 rounds total, less than a third of what the 262 carried) were origialy planned, but prooved to cause structural problems (due to the large muzzel blast causing damage, not due to recoil iirc) but this problem was solved before te end of the war, though no production machines were built as such.
The endurance down low is only ~30-40 min, (range being only around 300 mi) but this is much improoved at altitude, the 600+ mi figure being accurate with a max endurance of 85 min at 11,000 m in cmbat cruise of ~440 mph and a range of 100 km. This would be one major reason not to use it as a grount attack a/c.
See:
http://www.ww2aircraft.net/forum/stories/he-162-a-3234.html and:
http://www.ww2aircraft.net/forum/aviation/he-162-vs-meteor-mk-iii-2642.html
Also on the topic of fuel, Jets were actualy quite an advantage on this point compared to piston fighters, while they consumed much more fuel than prop fighters, they used J-2 fuel (basicly Deisel) which was still available in realitively large amounts in stockpiles in Germany throught the war and still at the war's end, while Galoline (particularly Avaition Gas) was in short supply. (the US Strategic Bombing Survey reporting over 100,000 tons of Deisel in reserve at the end of the war!) The main problem was that the German transportation network had been virtualy obliterated by Allied bombing, so getting the fuel to the planes was a major difficulty.
This fuel also has the advantage of poor flamabillity, meaning fuel-tank fires would be unlikely in combat. (although I wouldn't want to be in a jet with an engine fire/shot-up combustion chamber)