Yes, the penalty will follow the redirection
It won't help. In fact, the search results are so finicky right now
that you could probably redirect your domain to a competitor and get
them penalized. Basically, if your rankings dropped on April 23rd -
24th you're probably not coming back any time soon.
Check out this article by Barry Schwarz, and this comment by Google's Johnathan Mueller. Both address the what you should do in this situation, and how to submit a reconsideration request.
This article on how to avoid penalties is also worth a read (as are it's comments). The crux of the article is:-
Pay attention for an “Unnatural Links” notice in your Google
Webmaster Tools. If you get one, your rankings are doomed to slip if
you don’t do get those unnatural links to your site removed. It’s
important to note, though, that you may not get a notice before your
site takes a dive (or you may not get a warning at all), so don’t
assume you’ll be fine if you haven’t received a notice.
Make sure you don’t have any paid links. Buying links or selling
links that pass PageRank (meaning they aren’t NoFollow) is against
Google’s webmaster guidelines. If you are found to be paying for
links, or if it appears you’re paying for links, your site’s
rankings will suffer. To be safe rather than sorry, you should ask
for any paid links to be removed or made NoFollow. Contact the
linking site’s webmaster or customer service department and hope
Make sure you don’t have links from blog networks. Google has
cracked down on blog networks (like BuildMyRank), which are
typically basic-looking WordPress blogs with low-quality content and
keyword-infused links to other sites. For more information about
blog networks and Google’s update, read this article.
Sites with lots of links with keyword-rich anchor text look
suspicious. If the vast majority of the links to your site just
happen to use one of a few keyword phrases as the anchor text, they
aren’t going to look very natural to a person, nor to Google’s
algorithm. What are the odds someone would choose one of your top
keyword phrases when linking to you? Odds are, most people will use
something like your business name, the title of your blog post, your
business name, or “click here” as the anchor text when linking to
you. Make sure that the anchor text in your backlinks looks diverse
and not like you asked or paid people to give you links with
SEO-perfect anchor text.
Try to have a balance of high-quality and lower-quality links. Many
sites will have a few low quality links that they never asked for,
but it becomes a problem when the majority of your backlinks look
iffy. Look at the root domains (like example.com) that are linking
to you: What is their PageRank? Do they have a decent social media
following, or are many people sharing their content in social
networks? If most or all of the websites that link to you seem low
quality, you may be in trouble. It’s time to build up some quality
backlinks and/or try to get rid of some of the low quality links.
Avoid site-wide links. These don’t look too natural, and many sites
that sell links will put the links on all their site’s pages. See
the keyword-y links on the middle right of this site, for example.
There’s even an “add your link here” link right by them that goes to
a page where you can pay – DoFollow links costs 20% more than
NoFollow links, of course.