What it's trying to tell me --
N U T S
Put the answer words of each block "end to end", as it were, by making "some kind of connection"—a crossword-style connection, linking them at common letters. Correctly connecting the answer words will draw out a letter per block, and these put together spell out the four letter puzzle answer.
entries in BOLD were found independently by me (even though others got there too)
entries in BOLD ITALIC were borrowed from someone else (with credit given).
1. Talk to me, I talk back, ask me why, then I lie. PHONE (@Dan Russell)
you talk (in)to your phone and it talks back; phone-why→phony→lie
2. A fruit we are, like fat but not quite. PLUMS (@stack reader)
fat→plump→plum→plums (because "we" is plural)
3. I'm the happiness that's almost miss. BLISS
bliss is the happiness synonym that rhymes with miss
O U I
E S → "N"
1. Like someone on the warpath, just shorter. WRATH
someone on the warpath is mad as heck→wrathful→wrath; also, w
ar path leaves wrath
2. Rude drivers and angry bulls use me quite often. HORN
rude drivers honk their horns, angry bulls gore you with 'em.
3. Resting against a curve makes one tidy. CLEAN (@Thrax)
a curve, "C", with "lean" against it, makes clean=tidy
HORN → "U"
1. More like a cow than a cow, just scared. COWER
more cow-like→cow-er→cower, something you do when scared.
2. Rough weather or long road? WINDY (@Dreagen)
rough weather is windy. long roads can wind around = windy.
Y → "T"
1. We are shy, but when you're all gone the world is ours. MEEK (@TwoBitOperation)
the meek shall inherit the earth
2. Misleading acronym time? Step on it! MAT
[m]isleading [a]cronym [t]ime → mat
3. A most costly kind of gate. TOLL (@dcfyj)
toll gates cost money to pass; other gates generally don't.
4. Lay me down if you want to get your way. LAW
lay down the law to make sure people do things your way.
5. Put me on and do it good, I once got stopped by some big dude. SHOW
per OP, this is about WWE; Big Show's signature move was the "Show Stopper"
SHOW → "S"