Can there be a blockchain without mining? This is a question about Blockchain in general, not about Bitcoin as a currency on top of Blockchain technology. 

Let's say I want to issue a vote on top of, lets say, multichain. I would issue an asset called "voting_right". Let's say I have 1000 participants, so I create 1000 voting rights and send each participant address one voting right. Everyone can either transfer their vote to somebody else or directly to a vote option. 

I don't have given anyone mining permissions, because I don't want anyone to "illegally" earn voting rights. On the other side, I require miners to collect the votes into blocks and prevent double-voting. 


Can there be a Blockchain without miners?
If no, what I am assuming, is there any danger if I, let's say create two mining-enabled multichain nodes (with 0 reward), 100% controlled by me? How can others trust me?
I don't know anything about Multichain so I'm not sure if this is the answer you're looking for, but here's one answer that might help you.

The interesting thing about proof-of-work (Nakamoto) blockchains is that they enable a constantly-changing set of unknown nodes on the internet to achieve consensus under certain assumptions about the adversary (i.e., the good guys are more "powerful" than the bad guys). 

Before Bitcoin, no one knew how to do this (AFAIK) and all consensus algorithms required that the participants were known and that 2/3 of them were honest. Typically, an admission control process was used to decide who could participate and who couldn't. This was, for the most part, a human process.

Now, blockchains solve this very hard problem of permissionless consensus: How can you agree on a log of operations among N participants as they leave and join the system, without an admission control process? The answer: Use a little bit of economics to incentivize participants to behave correctly, no matter who they are (i.e., Bitcoin).

Now, with this understanding of the hard problem Nakamoto blockchains solve, you can image that a system such as the one you propose (if I understand it correctly), where users have to trust you to correctly dictate consensus, is not that interesting nor useful. It is well known how to achieve consensus with trust: just use any Byzantine agreement protocol and keep 2/3 of your nodes honest.

I am not sure if I properly understood your question and your system, so I hope this answer is not completely useless to you.