Yes, that is correct. You may not summon a pet if you already have a servitor up. The summoners servitor and the pets are both classified as "summons", and you can not have 2 summons up at once. Which, by the way, makes it a little difficult for a summoner to do the Sin-eater quest, since the sin-eater is a summon. You end up having to get friends to help.
As for the mount (strider) as far as I know that is a quest you can take to get one. You have to level it up alot before you can mount it though, lvl54-55 I believe. There are 3 types though they all look the same. Star, Wind and Dusk. I hope I got those correct.
Unfortunately for summoners, you can only have one summon OR one pet out OR be on one mount.
While there are a few other novelty pets (wolves, cougars, buffalos and kookaburras), only stryders and wyverns can be mounted upon.
Stryders are level 55 hatchings which have completed a special quest to become mountable stryders.
Wyverns are flying mounts which can only be mounted by clan leaders who own a castle or a few clan halls.
While stryders can give a very nice movement speed to whoever is mounting them, they also give very pathetic p.atk and m.atk since your weapon gets disarmed while you're mounted.
I own one mostly for looks or when I have to run quickly during raids and sieges to buff people all over the place, but that's about the only use you can get out of them.