a) If the nick is being opped by a special user that doesn't appear on the channel (e.g. ChanServ), you will need to make sure the allow-desync option is set to 1 in your bot's config file.
First of all, don't look at your config-file; use the .chaninfo command.
Secondly, nodesynch is a boolean channel setting; meaning it's either +nodesynch or -nodesynch, you don't set it to 1 or 0 or any other value.
Thirdly, when (if needed) altering any channel settings, use the .chanset command, don't edit the config-file unless you have full understanding of how the channels-file works (or maybe not even then).
On some odd setups, gaining channel op through server operator privileges, is seen as a server-op to other channel members. In these cases, you'll have to check the stopnethack-mode settings as well.
If you are using a bouncer, all your eggdrop should care about is that you've got a different hostmask.
If your eggdrop is using a bouncer, that would depend on what bouncer you are using, how it's configured, whether it (the bouncer!) does support (and use) any extensions/modules/scripts, etc.