As it was mentioned before, it works great on paper, but I have had issues myself getting Meraki and 3rd party (Even Cisco Catalysts) to play nicely in a mixed environment. Spanning Tree is going to be your enemy here unless you make sure your Non Meraki switches are set up to play nice in such an environment. Even then you'll run into situations where STP just freaks out and your network will have unexpected complications.
I personally have adopted stacking technologies as my standard best practice for enhanced redundancy to avoid STP issues. Stack your switches. Use LACP EtherChannels whenever possible. Use Layer 3 Segmentation as much as possible (Don't let your client VLANs on your Meraki's pass through your core - use a transit route between the MS210's and the Core). Try to limit your Layer 2 domains as much as you can -- don't let them span past switch stacks. This is especially important in a mixed vendor environment. Try to minimize your Spanning Tree as best as you can because that's where you're going to find the most problems. Unfortunately, STP is not an easy concept for a network newbie to grasp (I've been doing this 15+ years now and I still have a hard figuring out where the STP problems are).
If it was me, I'd virtually stack all of your Meraki switches together. All those MS210's will do virtual stacking so you don't have to have them adjacent to one another physically. If your "core" supports some form of stacking I'd enable it. Then use those cross-connected interfaces between your two core's (1 and 2) and MS210 (switch 3 and 4) as a 4x LACP channel. This will give you combined throughput (in theory) and sub 1ms failover redundancy.
Only thing you may find problematic is if you lose one of your MS210's. STP will try to reconverge again as the switch topology changes. Since they are Virtually stacked, they are still independent switches and have to learn the topology before they can finish the stack creation. You might find that an uplink port on your core goes into Error Disable during the reconvergence.
From your diagram though, using the ASA 5510 (might want to upgrade that EoL device!) is probably going to save you some of the headache I ran into with my Meraki MX firewalls... they don't do STP.