How to properly test office network quality?

My office's network is super slow when browsing HTTP and I am not sure what is wrong. If someone can point me the way to properly diagnose the problem it would be much appreciated. Some facts:

1. Old legacy network, probably 10-15 years old or so. Wirings are old, and could be chomped by rodents. Before my time.
2. Have a lot of 4-port and 8-port switches hiding under some nooks and cranny, and also two 24-port unmanaged switches. They are probably incorrectly connected topology-wise.
3. Main router is a Linksys E2500 running Tomato Shibby. This one I setup up myself.
4. ISP has 20 MBps bandwidth.

5. Very slow when browsing HTTP and email, frequently times out. Many times fails to even open main router's GUI page.

6. Great torrent speed! I got about 1.6-1.7 MegaBytes/s the other day, which is more or less what we pay for.
7. Many devices connected during the day. Can be 100 or so, including PCs, laptops, and mobile devices. Note that not all devices stay in office all day, and by afternoon probably only 40 or so devices are on the network. On a typical day total up and down bandwidth used is 5-10 GB.
8. Some ping results:

PING 192.168.1.1 (192.168.1.1): 56 data bytes

64 bytes from 192.168.1.1: icmp_seq=0 ttl=64 time=1.069 ms

64 bytes from 192.168.1.1: icmp_seq=1 ttl=64 time=1.242 ms

64 bytes from 192.168.1.1: icmp_seq=2 ttl=64 time=1.458 ms

64 bytes from 192.168.1.1: icmp_seq=3 ttl=64 time=1.510 ms

64 bytes from 192.168.1.1: icmp_seq=4 ttl=64 time=1.827 ms

64 bytes from 192.168.1.1: icmp_seq=5 ttl=64 time=2.495 ms

64 bytes from 192.168.1.1: icmp_seq=6 ttl=64 time=1.583 ms

64 bytes from 192.168.1.1: icmp_seq=7 ttl=64 time=1.393 ms

64 bytes from 192.168.1.1: icmp_seq=8 ttl=64 time=1.089 ms

64 bytes from 192.168.1.1: icmp_seq=9 ttl=64 time=1.375 ms

64 bytes from 192.168.1.1: icmp_seq=10 ttl=64 time=1.357 ms

64 bytes from 192.168.1.1: icmp_seq=11 ttl=64 time=1.266 ms

64 bytes from 192.168.1.1: icmp_seq=12 ttl=64 time=1.428 ms

64 bytes from 192.168.1.1: icmp_seq=13 ttl=64 time=106.366 ms

64 bytes from 192.168.1.1: icmp_seq=14 ttl=64 time=111.168 ms

64 bytes from 192.168.1.1: icmp_seq=15 ttl=64 time=66.989 ms

64 bytes from 192.168.1.1: icmp_seq=16 ttl=64 time=25.137 ms

64 bytes from 192.168.1.1: icmp_seq=17 ttl=64 time=1.436 ms

64 bytes from 192.168.1.1: icmp_seq=18 ttl=64 time=1.392 ms

64 bytes from 192.168.1.1: icmp_seq=19 ttl=64 time=1.341 ms

64 bytes from 192.168.1.1: icmp_seq=20 ttl=64 time=6.628 ms

64 bytes from 192.168.1.1: icmp_seq=21 ttl=64 time=1.694 ms

64 bytes from 192.168.1.1: icmp_seq=22 ttl=64 time=1.804 ms

64 bytes from 192.168.1.1: icmp_seq=23 ttl=64 time=2.421 ms

64 bytes from 192.168.1.1: icmp_seq=24 ttl=64 time=5.953 ms

64 bytes from 192.168.1.1: icmp_seq=25 ttl=64 time=1.351 ms

64 bytes from 192.168.1.1: icmp_seq=26 ttl=64 time=1.355 ms

64 bytes from 192.168.1.1: icmp_seq=27 ttl=64 time=15.504 ms

64 bytes from 192.168.1.1: icmp_seq=28 ttl=64 time=1.352 ms

64 bytes from 192.168.1.1: icmp_seq=29 ttl=64 time=5.576 ms

^C

--- 192.168.1.1 ping statistics ---

31 packets transmitted, 30 packets received, 3.2% packet loss

round-trip min/avg/max/stddev = 1.069/12.552/111.168/28.559 ms

PING (117.102.117.229): 56 data bytes 64 bytes from 117.102.117.229: icmp_seq=0 ttl=61 time=8.264 ms

64 bytes from 117.102.117.229: icmp_seq=1 ttl=61 time=7.649 ms

64 bytes from 117.102.117.229: icmp_seq=2 ttl=61 time=9.111 ms

64 bytes from 117.102.117.229: icmp_seq=3 ttl=61 time=102.421 ms

64 bytes from 117.102.117.229: icmp_seq=4 ttl=61 time=9.958 ms

64 bytes from 117.102.117.229: icmp_seq=5 ttl=61 time=12.041 ms

64 bytes from 117.102.117.229: icmp_seq=6 ttl=61 time=8.307 ms

64 bytes from 117.102.117.229: icmp_seq=7 ttl=61 time=10.437 ms

64 bytes from 117.102.117.229: icmp_seq=8 ttl=61 time=9.053 ms

64 bytes from 117.102.117.229: icmp_seq=9 ttl=61 time=8.305 ms

64 bytes from 117.102.117.229: icmp_seq=10 ttl=61 time=9.206 ms

64 bytes from 117.102.117.229: icmp_seq=11 ttl=61 time=8.594 ms

64 bytes from 117.102.117.229: icmp_seq=12 ttl=61 time=8.809 ms

64 bytes from 117.102.117.229: icmp_seq=13 ttl=61 time=7.019 ms

64 bytes from 117.102.117.229: icmp_seq=14 ttl=61 time=17.241 ms

64 bytes from 117.102.117.229: icmp_seq=15 ttl=61 time=40.051 ms

^C

--- ping statistics ---

16 packets transmitted, 16 packets received, 0.0% packet loss round-trip min/avg/max/stddev = 7.019/17.279/102.421/23.295 ms

Router stats:

Name TomatoUSB
Model Linksys E2500 v1.0
Chipset Broadcom BCM5357 chip rev 2 pkg 8
CPU Freq 300MHz
Flash Size 16MB
Time Wed, 13 Jan 2016 09:59:37 +0100
Uptime 0 days, 09:59:14
CPU Load (1 / 5 / 15 mins) 0.14 / 0.08 / 0.01
Total / Free Memory 59.96 MB / 40.43 MB (67.42%)
Total / Free NVRAM 60.00 KB / 35.05 KB (58.42%)

I have an engineering background and some programming experience, but I am by no means a qualified network engineer. Questions:

1. Is pinging an accurate representation of the quality of a network?
2. If I get rid of all switches and force all devices to use Wifi would it help?
3. Would having a Managed Switch plugged into the top of my network help?
4. Could Tomato firmware be a reason for high pings?

Thank you!

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