Dark Souls – Zweihander Sword Guide

Yes there are places that you can only get to by jumping. On the bright side however, they're not essential for progressing. Rather, they usually lead to some basic item or pick up. Off the top of my head, I've only discovered a light crossbow, bolts and one humanity in locations like this. Additionally, in most of these situations you can get past by rolling instead of jumping. From what I can remember, the bridge you're referring to is the most difficult jump I had to make.

The only other situation this is important, is done in order to return to the Undead Asylum (optional). You'll need to jump across to a ledge, but it can be done by simply rolling forward. Here is a video in case you're interested:

You are correct in saying that stat scaling doesn't really improve at all, beyond a minor improvement from ascensions (a sword might go from E --> D for example). In general, the stat classes that each weapon has in the game won't change much, even after being ascended fully.

Compared to Demon's Souls, upgrading a weapon to its max in Dark Souls provides less benefit because of this.

The bonus damage from stat scaling does increase for +n weapons, albeit it's a minor, linear increase (for example, a Battle Axe from +0 to +2 made its bonus damage from stats go +20->+22->+24). Thanks Tacroy

If you need to know when to replace your Drake's Sword, I would say get rid of it when you have high enough stats in one of the "damage" stats (Strength/Dexterity/Faith) and you have an adequate amount of resources to reinforce a scaling weapon to its max.