Your ASUS WL520GU is an ancient box that only does 100BASE-T (100 Mbit Ethernet) and 802.11g (54Mbps Wi-Fi, which will probably only give you 23Mbps in ideal conditions, and maybe 10-15Mbps in reasonably typical conditions). This is technology that hasn't been state-of-the-art since 2003, so don't be surprised that it's slow.
In contrast, your Netgear C3000 is based on 2007-era Wi-Fi technology. It can do up to 300Mbps Wi-Fi if you don't mind using band-hogging 40MHz-wide channels in 2.4GHz that cause problems for Bluetooth devices and Wiimotes and other users of the 2.4GHz spectrum. After overhead you should be able to see a little over 200Mbps out of it in ideal conditions, but in typical conditions, be happy if you're pushing 100Mbps. On the wired side, it has gigabit Ethernet, so its wired ports should be able to keep up with your residential broadband connection.
So, with the above info in mind, let's analyze your measured performance:
1) Netgear C3000 --> LAN --> PC = 220 Mbps
If you're paying for 220Mbps service from your ISP, this sounds about right.
2) Netgear C3000 --> Wi-Fi--> PC = 40 Mbps
It's a bit surprising this is so low, unless you're on a busy channel, you've turned off 40MHz-wide channels, you have a client that doesn't like 40MHz-wide channels in 2.4GHz, or you have a client that's only capable of single-stream 802.11n. In ideal conditions, this could be a little over 200Mbps.
3) Netgear C3000 --> LAN --> ASUS WL520GU --> LAN --> PC = 90 Mbps
This is expected since your ASUS box only does 100Mbit Ethernet. The theoretical max TCP/IPv4 throughput is barely 94 Mbps with standard-sized frames.
4) Netgear C3000 --> LAN --> ASUS WL520GU --> Wi-Fi--> PC = 10 Mbps
Pretty much expected for 802.11g in real world conditions. Picking a cleaner channel might get you a little closer to 20Mbps.
My overall advice to you is to stop playing around with retrocomputing. Your APs are based on 2003 and 2007 technology. In order to properly distribute your 220Mbps Internet service wirelessly around your house, you need the headroom that 2- or 3-stream 802.11ac brings you. So upgrade to 2013-era technology by buying a pair of APs that do "AC1750" or better. AC1750 means 3-stream 802.11ac in 5GHz for 1300 Mbps, plus simultaneously doing 3-stream 802.11n (N450) in 2.4GHz for 450Mbps there. And you don't just want AC1750, you want to make sure whatever AP you but has built-in high-power amplifiers. You'll have to read reviews and range shootout articles to know which have good amps.
Make sure to update your client wireless to 2- or 3-stream 802.11ac as well, because both the client and the AP have to support AC for you to use AC rates between that client and that AP.
I know it sounds weird to say "upgrade to 2013" in mid-2017, but the stuff that's come out since 2013 isn't that big a deal, and won't help you much. 4-stream 802.11ac is very rare on clients, so it doesn't help much to have it on your AP. So-called "Wave 2" 802.11ac with features like MU-MIMO doesn't really help much either. So-called "Tri-band" APs have a niche for users with a lot of slower 5GHz 802.11n gear around that they want to keep segregated from their fast 802.11ac gear, but that doesn't sound like you either. So you can stick with AC1750 or even 2-stream AC1200 if the AP has good high-power amplifiers. The next big jump in 802.11 speeds that will hit the mainstream will be 802.11ax, which I hear won't be finalized until 2019.