Modification to improve antibacterial activity
Essentially the basis of antibacterical activity of sulfonamides is the inhibition of  the enzyme dihydropteroate synthase, an
important enzyme needed for the biosynthesis of folic
acid derivatives and, ultimately, the thymidine required
for DNA.
Now they do this by competing at the active
site with p -aminobenzoic acid (PABA), a normal structural
component of folic acid derivatives.

PABA

So how does the structure of sulfonamide affect the bacterial
activity in general?

The basis of the structural resemblance of sulfonamides to PABA is clear. The
functional group that differs in the two molecules is the
carboxyl of PABA and the sulfonamide moiety of sulfanilamide.
Now there are requirements that are vital for activity:

Para-amino group:

unsubstituted (i.e $\ce{NH2}$).
very important for activity because any modification of it other than to
make prodrugs results in a loss of activity. For example, all
of the N4-acetylated metabolites of sulfonamide are inactive.

Sulfonamide:

$\ce{N}$ must be primary or secondary
$\ce{R2}$ can be substituted

Aromatic ring:

Para substituted

Modifications
The pKa of the carboxyl group of PABA is
approximately 4.9 thus an optimised sulfonamide has a pKa somewhere close to this value.

It was observed that replacement of one of the $\ce{NH2}$ hydrogens by
an electron-withdrawing heteroaromatic ring enhanced the acidity of
the remaining hydrogen and dramatically enhanced potency.
With suitable groups in place, the pKa is reduced to the same range as
that of PABA itself. Not only did this markedly increase the
antibacterial potency of the product, but it also dramatically
increased the water solubility under physiologic conditions.

Structural variation among the clinically useful sulfonamides
is restricted primarily to installation of various
heterocyclic aromatic substituents on the sulfonamide
nitrogen.
An example  is Sulfisoxazole (pKa 5.0) which can also be formulated as a prodrug, by inclusion of N-acetyl groups. Another advantage is the acetyl derivative
is tasteless and, therefore, suitable for oral administration, especially
in liquid preparations.


Is there an alternative for C as well?

Clavulanic acid is a naturally occurring, mechanism based
inhibitor , which causes potent and
progressive inactivation of β-lactamases.
There  are alternatives e.g sulbactam (A synthetic derivative of penicillins undergone the oxidation of the sulfur atom to a sulfone which greatly
enhances the potency) or tazobactam.
References

The 12th Edition of Wilson and Gisvold’s Textbook of Organic Medicinal and Pharmaceutical Chemistry
Foye’s Principles of Medicinal Chemistry 7th ed.