Also, I'm also curious as to why the engine favors 16.d4 or 16.Ba4 considerably more than 16.Nf5, which seems like the natural move. I can see why the first two would be candidates though.
Understanding the Ruy Lopez

i made a post regarding the d3 ruy lopez in the chessist society group...have a look...some proper annotated games by carlsen🙂
Cherub_Enjel (1076) vs. OPP (1200)
Let's Play | Chess.com | 6 May 2017 | ECO: C74 | 1-0
...
Are you the one who told us (about a day ago):
I've played ~2100 FIDEs, and ...
and
Literally all I did to get to 1600s from 1200 in a year was ...
?

1- The whole idea of the Ruy Lopez is to play d4 son. Im curious about what worries you, or why you would think it is not a good move. I see no way for Black to take advantage of d4, he is not in time.
2- No general principles here buddy, please. In this case I think d4 leaves you in a bad position, but there are many other positions where there is a bishop on b7 and White still plays b4.

No, i believe you have missunderstand the opening. The idea behind the Worral attack with 5. Qe2 is to push next d4 in case Black does not develop hes Bishop out of the pawn chain ( Bc5) as he does in the closed Ruy Lopez with Be7. After white play 6.c3 he is ready to support tactically and positionally d4 independence from castling and Rd1 and that he must do unless Black play Bc5. The idea for White to occupy the center early on is much more concrete.
Move orders are very importand. If Black doesnt play an early b5, white must consider even to give up the Bishop on c6.
In your question why 16.Ba4-16.Rb1 were better in your position instead of 16.Nf5 , the answer is that White is not ready to attack the Kingside when the exchanges favors Black there. After Blacks Bf6-Ne7 White has nothing better to do than to retreat the knight on g3.
The point of 16.Ba4 is to put pressure ( combined with Rb1 and d4-Qc4) on Blacks developmnent on the Queenside, when the Black Queen has to unpin going back , think which discoordinate the rest of pieces.
For more information about the Ruy Lopez and speciffic the Worral attack , you will find in Andrew Greet book : https://www.amazon.com/Play-Ruy-Lopez-Andrew-Greet/dp/1857444272

hari2017 @
Before you make assumptions about something, you have to understand by your own that the phrase : " When I played Rd1, preparing d4 the engine suggested d4 right away" is a post mortem analysis afterwards.
What's the point to use engine if you are gonna play inferior moves?? No need to be smart to understand.
Maybe try:
Starting Out: Ruy Lopez by John Shaw (2003)
https://web.archive.org/web/20140627024240/http://www.chesscafe.com/text/hansen53.pdf
The Ruy Lopez Explained by Gary Lane (2005)
https://web.archive.org/web/20140626201436/http://www.chesscafe.com/text/hansen80.pdf
Play the Ruy Lopez by Andrew Greet (2007)
https://web.archive.org/web/20140627023224/http://www.chesscafe.com/text/hansen94.pdf
http://www.jeremysilman.com/shop/pc/Play-The-Ruy-Lopez-76p3792.htm
The Ruy Lopez: Move by Move by Neil McDonald (2011)
https://web.archive.org/web/20140627022042/http://www.chesscafe.com/text/hansen153.pdf

I'm playing a bunch of random openings in my daily games, and the Ruy Lopez is one of them.
I'd like to learn more about this opening though.
(1) When I played Rd1, preparing d4, the engine suggested d4 right away. It's something I could play, but is a quick d4, when tactically safe, a good idea in such positions?
(2) Is my d3 move (instead of d4) correct, when black plays ...Bb7? This is to blunt the power of the bishop.
(3) Anything else you'd like to mention, especially strong players who are familiar with the Ruy Lopez?
d4 is ubiquitous in the Ruy... but substituting d3 gives a solid and low theory alternative. You see this structure a lot in high level play to avoid e.g. the Berlin or simply played on move 5. That said, it seems to be an odd mix (worrall with d3).
d3 is often the response to Bg4, not Bb7. Bg4 undermines white's control of d4 and with the pawn on d3 the bishop on g4 can be misplaced.
Anything else, hmm. There's a lot you can say about a few standard Ruy structures and ideas that apply to lots of different variations. I know some stuff, but not well enough to attempt to summarize it. I will say d5 is a big move for black. If black can get away with it, it's usually a big step towards equalizing. Since in many main lines white is preventing it one way or another, some players new to the Ruy can be somewhat surprised by this... even if they're playing the black side of it heh. Players on the black side who know very little usually move the c6 knight somewhere then play c5 never realizing white is allowing them to play d5.

Cherub_Enjel (1076) vs. OPP (1200)
Let's Play | Chess.com | 6 May 2017 | ECO: C74 | 1-0
...
Are you the one who told us (about a day ago):
I've played ~2100 FIDEs, and ...
and
Literally all I did to get to 1600s from 1200 in a year was ...
?
Yeah. It's a good thing I'm not 1200.
I'm playing a bunch of random openings in my daily games, and the Ruy Lopez is one of them.
I'd like to learn more about this opening though.
(1) When I played Rd1, preparing d4, the engine suggested d4 right away. It's something I could play, but is a quick d4, when tactically safe, a good idea in such positions?
(2) Is my d3 move (instead of d4) correct, when black plays ...Bb7? This is to blunt the power of the bishop.
(3) Anything else you'd like to mention, especially strong players who are familiar with the Ruy Lopez?