2 4, Could we implement some sort of HP-gain bottom cap so our characters don't roll 1 several times in a row? A lot of us are newbies, after all.
Some Homerules in my last game gave all starting chars +10 bonus Health, and I'm sure you start with your max HP at start.
Aside from that, one could also choose to just take average HP every level, or risk it with a roll. <--Edit: this was the rule in my last game, not standard from what I know.
There are a buttload of house rules relating to HP. One that I've used is that you reroll your entire HP every new level, taking the better of the two, your old one or the new one. For example, at Level 1 you roll 1d6 HP and add +1 per die for CON. You roll {6}, +1 per die, makes 7 HP. At Level 2 you roll 2d6 +2, and get {1,5} +2 means 8 HP. You compare the 7 HP you have to your new roll of 8 HP, and take the 8. This method effectively erases especially low HP rolls from earlier levels and you end up with higher HP scores on average. For monsters, roll the HP and then if the number is awfully low, reroll as if 1 HD lower and compare. This is largely the same result as if you had rolled for it as if it were 1 HD, then 2 HD, etc. up to its full HD. It means monster HP are also above average.
Generally, though, low HP aren't a problem. High stats can make up for poor play, and poor stats can add difficulty, but in general your player skill matters more. And: you know how you don't get better at something unless you do it? You don't become a good runner by driving everywhere? Playing a powerful character might stunt your development of player skill since you can rely on the character's strength instead of your player qualities. Losing a PC can be a good learning experience!
I guess what I'm saying is, if you roll 1 for HP at first level, it doesn't mean you can't survive or even flourish. Your careful decisions could save your life when you might just bust in there carelessly with a 10 HP character and get blasted.
EDIT: An early edition of D&D had a HP rule that might have been interepreted to mean that you reroll your HP before every adventure. You know how some days you're just bursting with energy and others you can't be assed to get out of bed?
The 1st edition Unearthed Arcana had a rule for minimum starting HP based on class, which prevented 1 HP PCs, and the rule may have made its way into later editions of D&D.
I've also done a training rule, where if you self-train for your new level you have to roll HP but if you pay for a trainer you get MAX HP. Training was pretty expensive though, 1500 GP x current level.
I find the main difference in games where people have average 3.5 HP per level, vs. 10 HP per level (and monsters are equivalent in any case) is that fights just take longer to reach the same result.