All classes can jump around, but Hunter has different jumps than Titans or Warlocks. Hunters have actual jumps and a teleport whereas Titans and Warlocks have jetpacks, basically. Most all classes can be very mobile if they specialize that way, but Hunters may have the slight edge with some of their abilities.
If you're looking for PVP weapons:
Primary: Most auto rifles are the same for legendaries stat-wise. Vanquishers from New Monarchy are the exception. Scout Rifles vary slightly but both Vanguard and Crucible have very good ones. Pulse rifles are under powered right now. Handcannons you need 9+ bullets, good stability, and a +range perk ideally. If you can get the pre-DLC stuff from vendors The Devil You Know from Vanguard is pretty awesome. Lord High Fixer from Crucible is very good as well.
Secondary: Shotguns for agile classes. I don't remember the pre-DLC shotguns but look for something with good impact, good range, and go to town. If you do it right most kills are 1-shot so you don't need quick reload or quick fire. Fusion rifles are good as well and 77 Wizard from Crucible rep is the best in the business pre-DLC and maybe beyond.
Heavy: Either look for a rocket launcher with Field Scout or 3 Tube perk. Then clusters on detonation ideally and seeking/early proximity secondary. If you want more kills for your heavy ammo get The Swarm machine gun (I forget which faction). Good rate of fire, good impact, decent stability, good clip size. Machine guns you want good impact, very good stability, and a large magazine. Everything else is a bonus. If you go high rate of fire and low impact you better have at least one stability perk on the guns.
Exotics for PVP: (I play on XB1 so I can't rate the PS exclusives, but they're apparently very, very good)
Primary: Thorn (Handcannon), Suros (Auto Rifle), The Last Word (Handcannon), Mida Multi Tool (Scout rifle), Red Death (Pulse rifle with healing).
Secondary: Invective (shotgun), Pocket Infinity (Fusion Rifle), Plan C (Fusion Rifle), Patience and Time (Sniper with Stealth built in), Icebreaker (Sniper with ammo regen)
Exotics: Gjalarhorn (Rocket Launcher). The other heavy exotics pre-DLC are not worth the exotic slot.
For PVE:
Primary: It's up to your play style. If you get an auto rifle get one with a huge ammo clip (field scout). Get a scout/pulse rifle with good stability. Get a handcannon with good stability and range with higher magazine. Don't use shotguns in PVE. They're not good.
Special : Use a fusion rifle for close up play. Charge time and magazine size are big for PVE. Impact is always going to be high, but never at the point where 4 in the mag is better than 6+ in the mag. Snipers for long distance you want something with Spray and Play for fast reload on empty clip, a magazine of at least 4, and firefly.
Heavy: A rocket launcher that blows up into clusters and holds as many in the magazine as you can find (2-3). Heavy Machine guns are basically all equivalent in PVE, just make sure they have good stability and decent magazine size.
Exotics: Suros Regime (Auto Rifle), Bad JuJu (Pulse Rifle), Pocket Infinity (Fusion Rifle), Icebreaker (Sniper rifle), Plan C (Fusion Rifle), Gjallahorn (Rocket Launcher) are all top-notch
EDIT
On DLC requirements: You don't need to get it to level. You can get to 28-30 pretty easily. You'll miss out on some stuff in the higher end because of their strike system using the DLC strike for weekly/nightfall sometimes. You'll also lose out on the easiest way to get good upgrades with their lvl 26 DLC strike playlist. But if you want to experience the first raid, most all the strikes, and plenty of PVP you're not missing out on much.
