Going in order...
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His mana issues early aren't that bad if you toggle Q on and off for a tick or two at a time. Most Singed players I see begin with Flask+3HP, so their mana is enough for some early farming. If you don't spam your E or leave your poison turned on too long, you shouldn't have many issues.
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His laning phase may be weak, but he's very powerful mid-to-late game if he can get his items through farming. In addition, Singed can proxy-farm waves to help by-pass his weak laning phase, since he can escape or suicide if enemies come to kill him and they lose out on gold and experience for the attempt.
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Most Singed players don't put a point in W until about level 8, at which point they should have a large enough mana pool for its occasional use (don't use it unnecessarily).
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His ultimate gives his a significant advantage in fights through extra movement speed, regeneration, damage, and durability and has a pretty long duration for it's cooldown. It might not be as instantly gratifying as many ultimates, but it's quite powerful. Also, his ultimate has a cooldown of 100s without CDR and a duration of 25s, so it doesn't have as much downtime as it appears.
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This can happen, but proxy farming can help you avoid a lot of bad matchups and apply a lot of map pressure when done properly.
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While Singed is quite item dependent, it's not because of his passive. Typically, you only want one mana item on Singed (commonly Seraph's Embrace, Rod of Ages, or Frozen Heart). While he does get health from mana, it's typically more efficient to just buy more defensive items after you have a sufficient mana pool.
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Killing people stupid enough to chase him is not the only purpose of Singed. He's capable of proxying multiple waves in the enemy base (allowing your team to push towers), splitpushing with a very low risk of death, quickly clearing waves to allow him to farm jungle or gank other lanes, and much more. Honestly, if it's a bad idea on most champions, Singed can probably get away with it.
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While his E range is pretty short, it has nice damage and a useful displacement which can easily catch people out, set up ganks, allow him to kite people, or make escapes. Singed isn't a great champion for towerdiving in general, until you get tanky enough that you can pretty much ignore tower damage and kill people anyways.
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Proxy farming allows Singed to largely avoid AP top laners. Even if he can't avoid them, he typically builds Mercury Treads and a health item (such as Rod of Ages or Rylai's Crystal Scepter) which can make it difficult for many AP top laners to kill him (especially through his ultimate). Some tankier DPS mages (Ryze, Cassiopeia, Swain) or champions with crowd control (Lissandra, Azir) can make it difficult for him to lane, but proxy is still an option. Also, while Thornmail is a very good item on Singed, it's not really required.
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Singed has high base MS, gets 35/50/80 addition MS on his ultimate, often takes Ghost, often builds Rylai's Crystal Scepter, has a strong AoE slow, and a displacement. While he doesn't have a gapcloser besides running quickly, he's very good at reaching a target and sticking onto them while being too tanky for them to kill.
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Eh, Singed has a pretty unique kit so he can't really be considered a typical tank or AP bruiser. Though, not fitting into a 'standard' type isn't really a weakness.
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Fun to play is entirely relative. While you may not enjoy playing Singed, that doesn't mean that other people don't enjoy his playstyle
tl;dr Proxy farming fixes a lot of his weaknesses in lane, his kit is very unique and doesn't easily fit into a standard archetype, his kit suits a gradual grinder (so it might not be the most immediately gratifying), he has a lot of options available to him which other champions can't safely perform, he's very good at sticking to people or escaping them, and fun is relative.
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), so he is super iffy to play in blind pick or as a first pick.
