OK so here is the deal, I have spoken with Google's John Mueller about this multiple times.

Your content will indeed be indexed, however Google bot is not as basic as many people think, there are multiple things happening when the bots come to look at your site, one of them is related to the humming bird algorithm. The bot takes a snapshot of what is actually visible on the page and will give THAT content more preference in search. It will also look what is above the fold for customers. So it is advisable to not have a large logo at the top and for some content to be visible without scrolling.

So just because some content is not visible does not mean it will not be indexed but it may be given far less weight than text that is immediately visible.

I hope that answers your question?

Those are know as "Sitelinks"

Here is the official page from Google about them, https://support.google.com/webmasters/answer/47334?hl=en

"At the moment, sitelinks are automated. We're always working to improve our sitelinks algorithms, and we may incorporate webmaster input in the future." Google

Having a good linking structure that is clear on your site, will help promote that. However you will also only get sitelinks when your site is an authoritative source for that search term.

Watermarking can have an advantage in Google images. If you have lots of images your brand will show up often and people will get to know it, without ever having visited your site.

Basically you need to be in Google News and then the article thats in the news must reach the top 10 in Search Results

Looks like this should cover all your questions

http://homebusiness.about.com/od/yourbusinesswebsite/a/google-alerts.htm

I actually learnt something myself as well on there, thanks for making me refresh my memory

yes screen readers use them but why not incorporate them in the site with css? It will help their rankings!

Rather than trying to get them to remove them, argue the seo case for them to show them.

they will get slapped at some point, if googlebot does not get them a disgruntled website owner that has a penalty will look at the top ten results, notice it and report them.

It is a direct violation of Google terms https://support.google.com/webmasters/answer/66353?hl=en

There is no problem in my opinion with using css to style your H1 tags so that they look nice on your site. But if you hide them you are just waiting for a penalty.

Also googlebot is much smarter than it was this time last year it is able to look at pages and see the content on the page rather than just looking at the code. this was part of the hummingbird release to target ad heavy sites above the fold but also looks for large logo's and content above the fold. All of this leads to the fact that Google knows what is visible and what is not and will rank you according to those factors. the H1 tags might be giving them no benefit at all right now without them realizing, When in fact they could be benefiting from them. how much business will they lose when they lose all their rankings for 3 weeks or maybe longer? Is it worth what they think they are gaining from it.

If they are really stubborn you could tell them to change a page or two and test it out. Do a fetch in WMT so the page is indexed quickly and see the results over the course of the week.

Its not a simple yes or no but, for the most part, redirects should not cause self-referrals

301/302 redirects should preserve the original referral information.

More info can be found on this exact matter on Google's site Under the anchor "Redirects and Self-referrals" it is right at the bottom under Common causes and solutions for self-referrals

https://support.google.com/analytics/answer/3198398?hl=en-GB

OK I think I found out some of your issues!

There are a few major ones. I hope you don't mind but I did some detective work and found out what your site was by the code screen grabs you posted on another reply below.

OK so the first MAJOR ISSUE is that you have TWO BODY tags in your HTML! You need to check your code on all pages / templates and get it fixed ASAP. Archive.com shows that this has been a problem on your site since you did a redesign in OCT 2013. Google is looking at signals like this a lot more now and is not being favourable to them

My concern when you do a site: query in google is that your homepage does not show, it should be the first result. I see that you have 15,100 pages indexed.

When I search for your website "sitename.com.ar" I see in Google there is no title, description even the quick links are broken, just showing just one and saying untitled.

NEXT, All your pages have the same description and keyword tags! Whats going on there? I you cant make them unique then get rid of them, they are doing more damage and Google hates that stuff, I have had pages not show in SERPS because they have duplicate description tags. This happen to five of my pages recently that I somehow missed. But this is on every product Page of yours if not every page on the site.

Also I would remove the keywords meta tag, its pointless and will do you more harm than good.

The paths to your javascript and css files have two slashes (//) should only be one.

I would start with those changes and do a fetch in Google WMT and give it a few days, the homepage should reindex in a very short time frame maybe minutes.

After that we can take another look at the site. These are very basic issues that a large brand site like this should never be doing wrong, as you can see the damage it has caused. Lucky for you that you have only just started there and can be the knight in shining armour to sort it all out