  By means of analyzing the notion of verbal products of groups, we show that
soficity, hyperlinearity, amenability, the Haagerup property, the Kazhdan's
property (T) and exactness are preserved under taking $k$-nilpotent products of
groups, while being orderable is not preserved. We also study these properties
for solvable and for Burnside products of groups. We then show that if two
discrete groups are sofic, or have the Haagerup property, their restricted
verbal wreath product arising from nilpotent, solvable and certain Burnside
products is also sofic or has the Haagerup property respectively. We also prove
related results for hyperlinear, linear sofic and weakly sofic approximations.
Finally, we give applications combining our work with the Shmelkin embedding to
show that certain quotients of free groups are sofic or have the Haagerup
property.
