Any wallet where I don't have to download the whole blockchain?
SPV wallets like Electrum, MultiBit don't download the entire Blockchain, just the block headers. This means that they do not verify the whole block but follows the longest chain that it can find. It presents a problem since it trust the miners and it was affected in the 4th July fork. However, this is a minor issue since most miners have stopped SPV mining.
What happened on 4th july? And what does SPV mining mean, like staking?
On July 4, majority of the blocks mined were V3 block and hence the network reaches consensus to switch to v3 blocks. On that day, a miner mined a v2 block which was considered invalid under v3 and hence miners rejected it and bitcoin Core V0.9.5+. However, miners running SPV mining continued on that wrong chain which resulted in those Bitcoins mined in those invalid blocks being invalid and there was a long block reorg for SPV clients(multibit/electrum etc) since it took time for big mining pools to switch to the correct fork.
No one reported any double spending attempts however. Miners SPV mine to eliminate verification time and thus reducing the stale rate.
So I only need to download the v3 blockchain, is that correct? If yes, what wallet should I use, an SPV or Electrum/Multibit?
No. SPV clients only verify the block headers. They trust the miners to enforce the rules and follow the longest chain. SPV clients are Electrum/Multibit. I prefer Electrum though. Anyway, this problem existed with Bitcoin Core below 0.9.5 and hence it's a must to upgrade the client regularly.
So the miners can manipulate the blockchain?
Thank you for you suggestion, I will stop my bitcoin core wallet because I dont want 60 gb to be wasted.
Full nodes verify each block to ensure that it follow the rules and consider them as invalid if they do. However, in the fork, clients running 0.9.4 and below thought that it was valid and proporgated it. Since SPV Clients do not verify the entire block, they thought it was valid too. I don't find it to be a huge drawback from using an SPV client unless you are a merchant which process large transactions on a daily basis.
Pretty normal situation for someone who wants to use Bitcoin Core. Meaning you need to download and sync all the blockchain information since the genisis block. AFter that, you have a running node and helping the network stay decentralized
If no one would have the bitcoin core wallet, then bitcoin would be a dead coin?
Or is the blockchain also hosted by companies, like blockchain, or is that different?
If no one is running a full node, Bitcoin would not exist. There must be at least one Bitcoin Core client running at all times. Services like Blockchain.info, Blocktrail.com all run a full node to keep track of the data.
.Good luck. 
