You could do so, but I highly recommend letting Marth fight. Usually a Lord becomes one of the best units after promotion (at lv 10 or higher). The way you plan your attack is up to you, completely.
You move your units as YOU want. Left, right, up, down, 2 panels or 4, attack or wait, use items, resque, take, blablabla. In FE9 you can even push one of your units to the next pannel. XD
No, the storyline doesn't change. But! Sometimes you need characters to recruit new units. For example: in FE 7 you need Florina to recruit her sister(s), you need Nino to recruit Jaffar (probably one of the best units in that game as I've heard), you need Matthew to recruit Guy, you need Priscilla to recruit Raven, in FE 8 you need a certain healer to recruit Joshua (another one of those deadly units) and so on....
So it's best if you keep everyone alive. =/