Barak,
I must say, at 24, you seem to be asking wise and intelligent questions. Most younger people, including me when I rode the TA at 17, didn't ask questions. They just think it would be fun to cross the country.
As regards to you questions. Yes, you can ride across the country with little to no experience. That said, the more you can ride before hand, the easier and enjoyable the trip will be. My classic example is when I was 17 and crossing the country for the 1st time (lots of riding experience though), I met two recently retired women in their mid-60s in Idaho along the route. The first had been cycling for decades and had no issues. The second had not cycled in 40 years, thought what her friend was doing sounded fun, and decided to join her. She went out and bought a bike, all the gear, literally rode around the block a few times to see if the bike was adjusted for her, then had it boxed up and shipped to Astoria. I asked her how the ride had been for her. She said the first two weeks were hell. But then her body adjusted and now she was fine and enjoying the ride. I figure if a 60+ year old person can do the route with no experience, then a young 24 person should be able to also. Granted, you might be sore for the first week or two but just plan on taking low mile days then. I don't know where you live in Israel, but I would when I was there a few years ago, I would have ridden most anywhere outside the major cities. Perhaps you can take weekends and ride out in the countryside?? I did ride a rent-a-bike along the shore path in Tel Aviv but that would get boring real quickly and would probably be just as dangerous with all the people on it.

.
The route can be ridden E>W from early May to mid-September and W>E from early June to late-October. The question is what is important to you, i.e. cooler/warmer temps, less traffic (end route in the fall but less services open, especially in the Rockies), etc. Riding E>W keep the sun at your back (important for safety if you ride when the sun is rising) but W>E has favorable winds in the high wind states like Wyoming. Other places, the winds is not as much of a factor. The extreme weather we are having is somewhat cyclical. I personally would not factor in the extreme climate as that is too unpredictable to forecast accurately.
The traffic on the TA is to a great extent relatively minor. Granted, cyclists do get killed occasionally but it is rare enough that it is fairly newsworthy when it happens. Use a mirror, ride with caution (no in the middle of the road), and be alert at all times (no ear buds playing music so you can't hear a car coming up on you) and you should be fine. I have tens of thousands of miles of touring experience and, knock on wood, I have not had an issue. The PC route has more traffic but also has more shoulders when needed so it too should be OK. If you do the PC, be SURE to ride N>S. I rode it S>N during a perimeter tour and it was not as fun due to the headwinds. See the sentence above about being young and not asking questions.
I also noticed that you posted a forum question over on CrazyGuy regarding meds. You did not say what meds, but I had a liver transplant and have to keep my meds at room temps or cooler. I basically just stuff them inside my sleeping bag and when I plan the packing for that particular route, I keep the sleeping bag on the "shaded" side of the bike if possible. I add a cool (not cold) bottle of water next to the pills on hot days. Remember, the pavement can easily be 30+ degrees F warmer than the air. I also pack each dosage if pills (I take multiple pills 3x daily) into little plastic bags (5cmx5cm) so if they do melt, they are just one big pill with the proper amounts of meds but that has never happened even in 105* temps. Again, knocking on wood, I have not had an issue in 15 years.
Tailwinds, John