Do you have any recommendations how Bloodseeker jungle can be countered? He's an insanely fast jungler and can just kill anybody with level 6 and some basic items.
Jungling, despite its popularity in pubs, is overall not a good place to put a core hero in from the start of the game. This is kind of a multifaceted issue that depends on the hero you pick, but in Bloodseeker's case it's just the most frequent answer of "it's just not as efficient as laning." Additionally, in Bloodseeker's case, the life return on kill doesn't give him perfect sustainability from the start of the game, which means that he will always have lower-than-full HP while farming. If you realize that the enemy Bloodseeker is jungling at the beginning of the game, you should see it as an opportunity to get free kills. Buy a smoke and run into the enemy jungle with another teammate. Some heroes could easily also solo-kill him (Clockwerk, for instance). Make sure that your teammates aren't at low HP or he might be able to escape or kill one of you, depending on what levels you are and whether he's skilled Thirst.
Is there any hero I can pick to punish his pick? Is there any hero that counters him and/or protects all my allies from him?
Is there any other item I should get against him?
Easy answer to this question is Omniknight, although that still does not counter his ultimate's damage. Be sure to stand next to BS so that he receives Degen Aura. Heaven's Halberd is a great item against him; wait until he uses Bloodrage on himself and then Halberd him, now he's taking extra-% damage for no benefit and can't attack you. Pugna is also a decent choice because of his Decrepify, and he scales decently; any hero comps that scale well into late-game will defeat Bloodseeker if their carry's gold gain isn't completely ruined. Coincidentally, me and a friend of mine think that 4-protect-1 will make a comeback in this meta. As usual against heroes that rely on movement and right click damage, Shiva's Guard is going to help a lot. Finally, this one is only half-serious, but if you find yourself the primary target of Rupture then Lotus Orb might help, since Bloodseeker really does not want to be Ruptured in a fight as a melee auto-attacker. I would consider that one a luxury. On that note, Rubick is a good pick against him; Bloodrage is also a useful spell to steal now that it also boosts magical damage, and since BS can only use either that or his ultimate, you're guarenteed to get something useful when you steal from him. Assuming that someone on your team can play him properly, then Tinker is probably also a good pick against him. Split-pushing is an excellent counter to gank teams as long as the split-pusher does not get ganked themselves, and Tinker is great at pushing while avoiding ganks -- the laser blind also helps against BS, as well as his incredibly fast farming ability, which leads into him buying a sheepstick.
Can you think of a way how I can play differently, so that he doesn't kill everybody on my team?
Pay attention to the minimap a lot, if you don't see Bloodseeker on it assume he's trying to kill you and/or anyone else. Always have wards up on the map. Always call missing. Don't push into enemy territory by yourself. These are the same things you should do against any gank-heavy strategy.
In a teamfight, you should try to spread out as a team. This makes it easier to kite melee heroes, and helps make sure you don't get hit as hard by AOE spells. Make sure your team has a Mek and make sure that it is being used before anyone breaches Thirst's health threshold in fights. The longer you can ensure that everyone has health above his Thirst threshold, the longer you also ensure that Bloodseeker has one skill slot that is doing literally nothing. Heroes that are good at healing can also help with this, obviously.
Before a teamfight starts, try to stay out of sight if you're a support or Int hero. You don't want to be the center of the Blood Rite. Concern yourself with avoiding the Blood Rite before you do anything else -- being silenced will make you useless in the fight after all.
If you have a strong slow, use it on Bloodseeker when the fight starts before he starts attacking anyone. The key is using it when BS's team (or yours) has committed to the fight, but before BS is in melee range of anyone. I would try to use your slow as soon as players are committed because BS will be gaining movement speed very quickly as people start getting low HP.
Another thing to consider about BS jungling is that it makes at least one of the other lanes on the enemy side weaker. Exploit the lane that is weakened due to BS's absence. Often in pubs, the enemy team isn't coordinated enough to pick an offlaner or proper hero that can solo whichever lane Bloodseeker is absent from.
Bonus Question: What are his weaknesses in the current patch/meta?
Since BS's ability Bloodrage was changed such that it is no longer a silence, it's pretty much never a good idea for him to use it on enemies now (having an ally deal 40% more damage is better than having one enemy take 40% more in almost every situation). Therefore you can consider BS as not having a silence any more, which means that he is not quite as good against caster-type heroes as he used to be, as long as they have some way to disable him. This means that you don't have to worry so badly about BS when you pick heroes like Crystal Maiden or Lion.
You still have to worry about Blood Rite, but obviously it isn't as reliable as his old silence. However, if you see Blood Rite being cast on you, you should assume that they have someone else there to disable you inside of it immediately.
Is there anything I can do in lane against him? I feel that the bonus dmg + heal is too strong to do anything against him as he can easily get every last hit.
One option is to use a hero with even stronger harassment. Good examples are Clinkz or Huskar. These heroes are really good at dealing damage to (especially melee) laning opponents. Another option is to deny his last hits; this can probably be done most easily with a hero like Timbersaw that strongly punishes farming melee heroes in lane. Be careful of being Ruptured as Timbersaw particularly, however. There are other heroes good at punishing melee heroes in lane; Lina, for instance, is very good at punishing melee heroes that go for a last hit. Watch your creeps' HP, and when one is about to die, time your Light Strike Array so that the enemy BS must be stunned if he comes close enough to last hit. Then you can follow up with auto attacks.
Being able to punish or prevent BS's last hits while maintaining a fair distance is probably the best way to outlane BS. Watch out for Blood Rite; if he casts it you should assume that a fight is about to start, most likely one with the odds not in your favor. The mid lane makes it fairly easy to avoid the AOE, though -- there are a lot of paths that branch away from mid. It's not an ability he can use to harass due to his bad mana pool and the cooldown. Beware of this ability if you decide to tower dive BS as well.