I will have (about to purchase) a 6" jointer with a 45" bed and a 13"
planer. I have a 10" contractor saw and a 10" chop saw.

The lumber yard seems to have 8' - 12' lengths and 12" widths, setting
skill level aside; can my equipment (I guess specifically my jointer)
handle these lengths? I'm thinking I'm going to have to cross cut the
boards to a more manageable length and then joint rip and plane.

Am I heading in the right direction?

Yes. And yes, your equipment is perfectly adequate for the task. I recently
upgraded from a 6 to an 8 inch jointer, but only because it was nice to have
both the extra width and length at times, not not necessary

Yes, crosscut to 1" more than you need to start. If, for instance, you had
an 8' board and needed two 4's, if you joint then crosscut, you will loose a
lot of material flattening any bow, and secondly it's pretty hard to joint
an 8' board on a 45" bed and have it come out really flat. So crosscut
first.

If you have pieces wide than 6" rip slightly oversize on the bandsaw. If you
do not have a BS, edge joint the board before ripping on the table saw (it
makes ripping a rough-cut board a little safer). Then go back and face/edge
joint

Rip 3rd edge on the TS, then plane opposite face (or "vise" vera)

If you want to end up with a wide board (e.g. table top), you are going to
end up gluing it up out of multiple 6" boards anyway. Sometimes you may,
however, want a a solid 10" board for something like a big drawer front. In
that case, you can use some combination of face jointing by hand and light
passes in the planer to get things pretty flat.

6" is not that much of a limitation at all. You have the right toys.

-Steve