I am currently trying to get more into sharding but there's one question I haven't found any answer to yet.
Could sharding introduce shards with different properties? For example could Shard A have a block time of 10 seconds, a gas limit of 8 million and Shard B a block time of 4 seconds and a gas limit of 2 million?
Yes, this is possible, but it would be more complicated to implement.
See also https://github.com/ethereum/wiki/wiki/Sharding-FAQs#what-about-heterogeneous-sharding.
What about heterogeneous sharding?
Abstracting the execution engine or allowing multiple execution engines to exist results in being able to have a different execution engine for each shard. Due to Casper CBC being able to explore the full tradeoff triangle, it is possible to alter the parameters of the consensus engine for each shard to be at any point of the triangle. However, CBC Casper has not been implemented yet, and heterogeneous sharding is nothing more than an idea at this stage; the specifics of how it would work has not been designed nor implemented. Some shards could be optimized to have fast finality and high throughput, which is important for applications such as EFTPOS transactions, while maybe most could have a moderate or reasonable amount each of finality, throughput and decentralization (number of validating nodes), and applications that are prone to a high fault rate and thus require high security, such as torrent networks, privacy focused email like Proton mail, etc., could optimize for a high decentralization, low finality and high throughput, etc. See also https://twitter.com/VladZamfir/status/932320997021171712 and https://ethresear.ch/t/heterogeneous-sharding/1979/2.
