If I have an interface that is in VLAN1 (default VLAN) then tag it in VLAN 20 it is automatically removed from VLAN 1 altogether. This behavior is very strange to me. To place it back into VLAN 1 as untagged I am forced to configure it in VLAN 1 as dual-mode. Is there a way around this?
No, it's correct.
Generally speaking, VLAN is ment to be used in the "distribution" side, ie. between switches, and usually you just want to carry only tagged traffic; untagged frames in this context is just signalling and control (say, STP BPDU and such)
but it is not traffic that is ment to be forwarded to other ports.
On the opposite side, to the terminal side the computers and such won't understand VLAN tag, and you should put each port in its network, untagged of course.
The exceptions is covered via the dual mode property (say, for VoIP phones), and in this case best practices suggest to avoid VLAN 1 altoghether.
I can have a Brocade SFP+ Transceiver plugged into interface 1/3/4 that is working without issues, but then move it to interface 1/3/3 and it won't detect it.
It's harder to troubleshoot with the breakout cables as I can only swap an entire breakout cable at a time and I only have two breakout cables, but the results appear to be similar.
That is strange at best... it could be licensing related (but it shoud enable contiguous block of ports, and you can see in that case on a show interface 1/3/3 that it is err-disabled), or it could be defective... Try posting a show interface ethe 1/3/3, show licence, show pod and show optics when the module is plugged in the 1/3/3 if.
EDIT: just to be safe... are you sure the interfaces aren't DISabled in their configuration? Could you post also a show running-config interface eth 1/3/3? Thanks