Different algos are based on different programs. Different algos have different proposes and each has its pros and cons. I believe anyone can invent an algo but to have it work and be well adopted is the challenge. Best example one can relate would be Charlie Lee invent scrypt algo which mines coins such as Litecoin.
Your last question is truly a cause for concern. The only way someone can rewrite the previous block is if they control the block before that. My point is if someone is mining, or hashing out all the blocks they essentially have full control to change any of the blocks. Here is were decentralization comes into play. Multiple people over the globe can be hashing the same blockchain. The hashing will break out who gets to right to approve each block and each block would be a different person as there will be multiple people hashing. The more hashing the faster someone gets to approving a block. A side bar you should look at 51% mining attack. When one entity has the marjority of hashing on the blockchain they can essentially corner the blockchain and create every block as their own because once they approved a block the next block can be again mined by them because they have the majority of the hashing. This is why it is so important for a coin to be adopted by all that the mining be well spread all across the globe. Thanks.
Even though I already knew this information I found your explanation to @va2099 very insightful, please keep up the good work with your curation...