A lead sheet tells you next to nothing about when to play a chord. You can play the C chord on the downbeat, or you can play a low C on the downbeat and the chord on beat three, or you can play a low G on the downbeat, the chord on beat two, and a low C on beat three, and the chord again on beat 4, or you can play any of dozens of other possibilities.
The rests in the melody belong to the melody only. The rhythm of the accompaniment is at the discretion of the people playing the accompaniment.
In a comment, you ask
What does is mean to "play a low C" - a single press only on the C key?
There are several C keys. "Low C" typically means the one that is two octaves below middle C. And the answer to that question is "yes."
What I am trying to convey is that any of the following examples would be acceptable (though the F on beat three of the second measure is a bit odd with the C chord, so one of them might be wrong -- if not, you may want to choose a figuration that minimizes the clash such as the second example).
X:1
%%score { RH | LH }
L:1/4
V:RH clef=treble
V:LH clef=bass
K:C
V:RH
CCCA|zCFC|
V:LH
F,,2[C,F,A,]2|C,2[C,E,G,]2|
V:RH
CCCA|zCFC|
V:LH
F,,[C,F,A,]C,,[C,F,A,]|C,,[C,E,G,]G,,[C,E,G,]|
V:RH
CCCA|zCFC|
V:LH
F,,[C,F,A,]C,,[C,F,A,]|C,,[C,E,G,]G,,[C,E,G,]|
V:RH
CCCA|zCFC|
V:LH
L:1/8
F,,A,,C,A,, F,,A,,C,A,,|C,C,,E,,G,, C,G,,E,,C,,|
V:RH
L:1/4
CCCA|zCFC|
V:LH
L:1/8
F,,>C,D,>C, F,,>C,D,>C,|C,,>G,,A,,>G,, C,,>G,,A,,>G,,|
V:RH
L:1/4
[F,A,C][F,A,C][F,A,C][CFA]|z[E,G,C][G,CEF][E,G,C]|
V:LH
L:1/8
F,,>C,D,>C, F,,>C,D,>C,|C,,>G,,A,,>G,, C,,>G,,A,,>G,,|
In a comment, Todd Wilcox wrote
IMHO if the lead sheet writer doesn't want the chord to be played on the first beat, they should offset the chord symbol to line up with where they want it played.
Many lead sheets do employ this degree of precision in the placement of chord symbols. Others don't. (Similarly, some are more precise than others about specifying the bass line with slash chords.) Over time, you will learn to judge what the author of the lead sheet is trying to convey and how much liberty you can take with their prescriptions.
But keep in mind: Inherently, the very purpose of a lead sheet is to leave the details to the performer. The lead sheet is the wrong tool for a composer or arranger who wants a specific rhythm in an accompaniment; the right tool for that is staff notation.