Signing into alternate

Did you check the Spam folder for your alternate email address? Can you confirm that by going to account.live.com, sign in there (you will be able to sign in even with an Xbox account problem) then choose Edit security info to look at the alternate
email address.

is your profile on another xbox? is it possible someone is signing you in on another xbox as this would sign you out elsewhere.

if not maybe try clear your alternate mac address and see if it helps. settings-network settings-advanced-alternate mac-clear

Have you tried to sign in at account.live.com and change the alternate or alias on the account?

Can you delete your alternate email address and rename your primary to that one, or can you rename your primary to your alternate and rename your alternate to your primary then delete your alternate (which is the primary turned into the alternate) (sorry
if it's confusing), while you are entering the verification codes isn't there a little box that says "i sign in frequently on this device don't ask me for a code" if so click it, as for the Xbox console sign in part, here is something i found,

Re: Email and Password prompt every time I try to sign in! - 03 - Managing my Xbox LIVE account - Xbox Support - Xbox.com Forums - Xbox.com

@ Dark - I haven't actually bought a new GPU yet. Just going through my options right now. Had to buy a PC NOW with school getting ready to start. So I went out and found the best deal I could with the budget I had. I'm happy with the results. Got a
lot of PC for little cash. The cool thing is that the new PC was a necessity... the PS4/X1 is a convenience... if I really want to I could drop $400-$500 on a new GPU, as that's what I'd spend on a console... but at the same time, $150-$250 will still offer
a huge upgrade and be more than sufficient to meet my needs.

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Good points you're making though. There is give and take with PC vs console (exclusives, graphics, online, etc), but some things that have swayed me are: cheaper gaming (Steam sales, free to play, etc), less restrictive gaming (mods, indies, etc) and the
ability to upgrade the system in the future, as I see fit.

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I agree on the optimized comment, but not on "everything a PC can do an XBone or PS4 can do". It's because consoles have such limited applications that they're able to be optimized and compete with the PC market for 8-10 years per generation. They play
games and run some limited apps/content. Sure, more processing power might get squeezed out when similar hardware is compared side by side... but consoles just don't have the flexibility or massive multitasking capability of PCs.

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I'm not saying I'll never own a PS4 or X1... the PS4 is more likely, and the X1 is less likely (unless the kinect gets removed, but let's not go down that road, we all have differing opinions on it, and are unlikely to change). So eventually I might get
around to playing some console exclusives. But there is plenty coming out in the future on PC that will keep my interest - BF4, Elder Scrolls Online, Titanfall (huge one for me), Batman: Arkham Origins, Assassins Creed 4, etc.

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And it's entirely arguable that console is better for gameplay... PC's often offer more in depth controls and settings that a console just can't.