Easy part first: Yes, it is possible to have every class maxed on one character.

For cross class skills, it's never necessary to take the sub classes to max level. Only certain skills can be cross-classed. For most classes, the highest skill comes in around level 34 (there are exceptions).

Here's a reference for the possible skills for a Monk (and you can use it to check out the skill trees for any other classes you're interested in). Blood from Blood from LNC is a level 34 skill. Mercy Stroke from MRD is a level 26 skill. There are no cross-class-able skills for either class after those two. And, in this case, those are both skills you'll eventually want.

It's not necessary to get them right now, though. You don't really need your full range of cross-class choices until you're doing end-game content, and even then, it really depends on the class and the content. Missing Blood for Blood and Mercy Stroke isn't going to lock you out of much before Extreme Mode primals and Binding Coil of Bahamut (the current top-end raid). I would suggest getting Invigorate (LNC 22) before you go too far into level 50 content, but it's not that vital before then.