Technically you're always at the mercy of your ISP. Each ISP does their own thing, and their WAN DHCP systems can also work very differently from what you'd expect from a "normal" DHCP server.
2) What controls over the DHCP lease do I have as a DHCP client (e.g. am I able to request for a new IP or am I at the mercy of the DHCP server?)?
It depends on several things, e.g. how the server stores leases in its database:
Some DHCP servers implement "sticky" leases: they remember your lease in a database even after it expires and/or even if it's deliberately released, so you still get the same address on new requests.
Other DHCP servers only let you keep renewing a current lease infinitely, but as soon as it expires/gets released, it's gone – new requests pick an address from the available pool again.
(Some ISPs don't even allow renewal for more than X days, forcing you to get a new address every time. This still falls under "as soon as it expires it's gone".)
If you're renewing a 'current' lease, or if your ISP implements sticky leases, then it also depends on how the server identifies you as "still the same client".
Standard basic DHCP identifies clients by their "Client ID", or lacking that, by their MAC address. Different MAC address will get a different lease, but the same client moved to a different location will carry its address with it.
Some ISPs configure their DHCP server to identify clients by their physical connection (option 82), so you get the same address even if you replace the entire router.
1) With regards to DHCP renewal, do i have any control over me receiving a new IP or am i at the mercy of my ISP?
The button named "Renew" actually means "Do whatever you must to acquire an address, renewing the current lease if possible, but getting a new one if that fails."
Renewing a lease that you currently hold – if it hasn't expired yet – is supposed to keep exactly the same address as the lease originally had. After all, the "address lease" is literally a permission to use that specific IP address.
(This means a normal renewal doesn't cause any network disruption – not even for a millisecond. A host might keep renewing its address every hour for years, while still maintaining the same connections.)
But if you get a different address, this means the renewal failed and your router had to start the process from zero again, getting a new lease for you.
This might happen if the old lease actually expired (i.e. the router didn't automatically renew it in time). But it can also happen because some ISPs deliberately configure their DHCP servers to deny renewal after the original lease duration, to force you into getting a brand new lease every day.
Performing a release and renew will eventually have the same effect. To the router it "renews" the lease.
No, it's not the same thing from the DHCP server's perspective. The "release" operation doesn't just stop the process, it actually sends an indication to the server that it is now free to discard your old lease. It only so happens that some servers continue giving you new leases with the same address, but it's not required to.
"Renew" performs the opposite steps: first it specifically asks the server to re-issue the same address, and only if the server refuses, then your router asks for a brand new lease.
