Is it ethical to request the advisor to share the registration fee for submitting a conference paper?
Talk to your advisor, now.  Dealing with funding and finances are a significant part of academic life; they aren't some shameful taboo thing.

Most academics don't pay for professional travel, including conferences, out of their own pockets; often funds are available from your institution, or from any grants that may be funding your project, or from the conference itself.  It's entirely reasonable to ask your advisor about funding sources that might be able to pay the cost for you attending this conference; you should also research possible funding sources yourself.

If funding isn't available, or can't cover the full cost, it's up to you to decide whether you want to spend your own money to go.  If you decide that you can't, that's okay; you just won't go.  I don't think people would see this as stinginess on your part, but rather a misfortune that no other funds were available.

However, by the same token, if your advisor tells you that he doesn't have any funds available to pay, you shouldn't expect him to dip into his own pocket.  

Ideas like your  "50/50 split" don't usually come up, unless there are multiple outside funding sources that could pay, and the question is how to apportion the costs between them.