I rode most of the horses at some point, pre and post patch, using a 1h weapon+shield. My favorite horse both times was a Sarranid Horse. As for weapon - Long Arming Sword ( any weapon with over 100 length will do. Nordic Champion's Sword/Sarranid Cavalry Sword would be my first choice, if i was only going after "usability", Arming Sword just looks better for my taste. Had a masterwork "edition" before the patch... ) Fighting Axe can be fun too, having 8 PS pre-patch, i could trash some shields in one charge...
If your gear is expensive ( mine is ) - a Steppe Horse will do just fine. Put more points in Riding - it will give your Steppe horse more speed ( so it was pre-patch at least ). Courser is a fun horse to ride, but pure speed is not so important to me, as i often have to fight on hills, in villages, and even on castle walls...
Sarranid Horse is The Queen of the post-patch cRPG! With light/cloth armor, you should be able to support the upkeep.
Stay away from anything that can shoot at you. 2-3 arrows, 1-2 bolts or a single throwing axe - will kill your precious pony... Take a light crossbow as a backup weapon - enemy cavalry will ALLWAYS mark you as their top target, since you are using 1h weapon, and they have lances. Point your crossbow, and see how they make a U turn.
When mounted, the best i can do for my team as a 1h user - assist in small fights, between 2-5 people. Pole-horsemen need to gain speed and some distance to be most effective, while 1h weapon can be used while being almost stationary. So I can harass the enemy from the back in close range, making them an easy target for others. Versus enemy cavalry - 1h suck.
EDIT: I agree with Trikipum to some extent, I rarely use my horse in the patched cRPG... Even charging a 1h+shield enemy is too risky now, i mostly chop reloading crossbowmen... They always remind me a shitting dog, same position, and they look around like crazy, knowing their "vulnerable position", haha.