Purpose of this diagonal board on underside of deck

The rule, at least here in the US, is a cantilever can extend no more than 1/3 the supported length (so, you could go ~3 feet further). Personally, I try to keep cantilevers no more than 2 feet, regardless of total supported length: while it might be OK structurally, it will likely sag the further you go, and can feel bouncy.

To do the best job, get some 12 foot 2x8s and sister them with the existing joists (run them the entire length, nailing with galvanized nails at the top and bottom every 16 inches). Then put a double joist hanger at the ledger against the house.

One of the critical things with a deck cantilever is how you attach the railing posts at the end of the cantilever. You will be nailing a 2x8 rim joist at the end to finish the deck, but you need to make sure the posts for your railing are tied in securely to the joists, and not just the rim joist. The method can vary depending on whether your existing railing posts are notched into the decking or fastened to the outside of the deck.

I had a deck where the rim joist bowed out, and the cross joists came unattached. I used two car jacks to get everything back up to level to be reassembled and use proper joist hangers.

Safety tip, use the jacks to lift, but once up, set a stack of lumber or blocks to hold it up while working underneath. If you have some 4x4 stock (or 2 2x4s edgewise), don't lift a specific joist, but use a cross piece to lift several and spread the force.