Why I struggle to move my Queen sideway?

Let's say in the puzzle below

I fail to see that my queen could actually go sideway. 

Again here

Which is a good example of how queen can move sideway. 

What thinking process I have to change to get it so that I could see it intuitively? 

I used to have the same problem. You just have to keep reminding yourself to look for it and eventually it will just happen.

How about this one: Is 6. Nxe5 a good move, or a blunder (assume that Black will reply with the correct response, 6. ... Nxe5 intending 7. ... Nxc4, not the error 6. ... Bxd1 getting mated):

Yes, White has Qb5 at the end of the variation. I solved it without moving a single move. For me White probably realise Nxe5 by white is a mistake since black doesn't get mated. To avoid loss of material I found Qb5. Phew! 

Correct.

After 6. Nxe5 Nxe5 7. Qxh5 Nxc4 8. Qb5+ followed by 9. Qxc4 White has won the e-Pawn for free.

IkR

That was one of my over-the-board tournament games against a 1900 player.

It is a common problem you describe there. You should take a look at this one:

https://www.newinchess.com/invisible-chess-moves

It's common to miss lateral movements like that.

I've done tricks in blitz where I put my queen on a4 to threaten h4 (just a random example) and I always think to myself, heh, he's going to not see it.

Getting a cheap win because he doesn't know the variation of legal mate. Ay.  But I must agree one pawn advantage is difficult to convert against an 1900

It's not only a pawn, it's also development and the center in a calm position.

It's not guaranteed a win, but it will be extremely hard for white to lose.

And 1900 might be hard for you, but 1900 is not hard for 1900

LMAO

I purchased game changer book recently at forward chess. I trying to reach as far as I can go by playing more than 45 min per side. I am reading the first chapter. So,  those chapter are still some far ahead. 

Take all the pieces off a regular, real chess board and practice all the pieces moves. Then the simple endgame mates if you want.  It's a matter of seeing the Queen move in that way over and over... and reminding yourself to look. (This is one thing Bruce Pandolfini taught me when he introduced me to chess [he was a neighbor in NYC]).

You need to take almost everything off the board and practice mating with the Queen and another piece, for example. 

Train your memory to look for those patterns. 

(Bruce taught me two things. I didn't play at the time. That's what got me interested in chess.)

I play puzzle rush battle. Is that will do for pattern? I used to score 18 regularly in that. 

If you're playing puzzles, the best thing to do is practice tactics puzzles. (My puzzle rating on here is low because I couldn't see the puzzle theme on my app version of chess.com, and I'm fairly new at studying chess in depth so surely would have helped on Learning Mode).

I did practice tactics like crazy on Chess King apps (rated 1649 for puzzles)  and that helped me recognize some patterns for sure. 

Also remember that it's normal for humans to look at things linearly, not on diagonal--except we know the bishop only moves that way, so knowing that, that's what we look for. 

Again, I'm no pro (Ha! Far from it!), but I'm learning like you and I think practicing with the pieces and doing tactics puzzles will help you. 

I'm sure much more seasoned players can give you more specific help. I hope what I've offered helps you a bit. 

It's like muscle memory when playing a musical instrument, but here you have to train you eye to see the patterns.

You can do it! 

Even more common is to miss backwards moves.  Take the following diagram.  It is Black to move.  What should White do if Black plays 1...Be7, threatening the Knight with check?

I am a defensive guy. I can see queen go back. 

Even if i cheated  by reading your prose. 

Yes, but you are not moving the Queen back for defensive purposes.  You are moving it backwards on the second move of a mate in 4, after sacrificing the Rook.