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I have built a toroidal inductor by wrapping an insulated copper wire over a toroidal laminated soft-iron core. The dimensions of core and copper wire are as below:
Toroidal core: Inner diameter = 20 mm Outer diameter = 70 mm Height = 15 mm
Copper wire: Total length = 150 m Diameter = 0.100 mm
My main concern is that what maximum DC current and voltage can I allow to flow through this inductor?
Your photo shows a closed torroidal core. All the flux will be contained inside the core so this will not perform as an electromagnet as you have supposed in your question title and post.
Core diameter will not affect the flow of current (except at switch on due to inductance). Your problem is simply one of wire resistance. See Current capacity of very thin (0.1mm ~ 0.5mm) copper wires? for details on that.
What maximum DC current and maximum voltage can I flow through an electromagnet?
Current flows through. Voltage is applied to the load. Once you have calculated the resistance of the wire and decided on the current required then you can calculate the required voltage from Ohm's law, V = IR.
My main concern is that what maximum DC current and voltage can I allow to flow through this electromagnet?
The main question is 'for how long?'
The failure mode of that copper wire is it getting too hot for the insulation. If you want to run it for tens of minutes, then the heat must be dissipated continuously. While it's theoretically possible to compute a temperature rise from dimensions and fluid mechanics of air, it's far easier to just switch it on at low power and see how hot it gets. Rinse and repeat with increased power until you just can't touch the coils (will be 60 to 80 °C) which most insulation should survive.
If you want to run it for seconds or milliseconds, then heat will be absorbed into the thermal capacity of the wire rather than dissipated, and you will be able to run very large currents, with a constant value for \$I^2t\$. Reduce the operating time by 4, and you can double the current. Pulse it on for a second, estimate the temperature rise, and use \$I^2t\$ to extrapolate to other times and currents.
BTW, what you have there is not an electromagnet, it's an inductor with approximately zero external field (exactly zero if the windings are uniform). It won't attract pieces of iron, and should barely affect a compass needle.
