Chase may pull EQ and EX. Nothing wrong with apping for both same day. I'd hit Chase first. Any other cards you want with Chase? I'd suggest you two for one when possible. Amex doesn't care about inqs or new accounts.
Well depending on who you fly with or hotels you use. Chase may have a card that will make the flights free or the hotel stays free. I would go over to the site and look through what they have to offer. In the future you could move the limits to your preferred card or cards and close the account.
Yes it's only one set of pulls if done on the same day. App using two different browsers.(for example Chrome and Explorer)Wait at least two minutes and then do second app.
You could also clear browser history and use the same browser. I've seen Chase's system get confused if you don't use a different browser or clear the browser history. The second app won't register and present the same screen of approval or processing lingo from the first app.