Right, but generally I just like to install and play :)

Well, Mass Effect is the only (and last...) game I've bought from EAStore, so I don't know what other games need that download manager program. Some retail games can be linked to the eadm (if it's EA's game and quite recent one) and that can be played without cd using the EADM. But it needs to be online (at least I don't have the option to even browse my game library offline, since all it complains that it cannot find internet connection....
In general, if you buy something from EAStore, it uses the EADM to download itself. Then it's a pretty good chance that it needs the EADM to work :)

I don't think any games require the EADM thing... The Sims 3 maybe, I don't have that game and I know it has extensive in-game store stuff, but none of my EA games require it.
The Dragon Age thing is different, it's a unique program called something like 'Dragon Age File Manager' or what-not. It's there because unlike Mass Effect 2 Dragon Age handles DLC downloads and authorization in-game.

Without more information, nobody can help you.
That being said, on my older win7 x64 machine (more than a few years old :p) I could never get Mass Effect to run. Would crash out when it tried to load the splash screen.
But on my new win7 x64 machine, MEPC works perfectly. So go figure.

I'm using Windows XP, the same machine I had when I finished Mass Effect (Steam version) on normal difficulty some time ago. I was gonna start a hardcore run yesterday and for the first time since I bought it ME crashed within the first minutes of playing. It happened twice and the only thing I changed since last time was that I forced 4x AA on nVidia's control panel. I didn't have time to test if disabling it stopped the crashed but I hope so, otherwise my PC may be dying. The funny thing is that I have forced AA on ME2 for a long time and it didn't crash ever (unless it was a known crash like the one in the Overlord DLC).

Mass Effect is one of those games like Fallout 3 that is very sensitive to driver issues. The wrong kind of printer driver or whatever can make the game crash consistently. For this reason among others I tend to run a clean PC for gaming with nothing installed on it but essentials and I leave the printing and such for my laptop.
I never, seriously never ever, have crashing problems with games. Ever.