Relation between IVT, Connectedness, Completness in Metric Space, LUB

The motive behind this question is:

1. In an ordered field LUB <=> IVT.
   Is the IVT equivalent to completeness?

2. Connected Metric Space X does have IVP i.e "all continuous function f:M⟶R that admit a positive value
   and a negative value, also admit a c∈M such that f(c)=0".
   I read that the converse is also true.
   If M a metric space with the property of Intermediate Value. Show that M is connected.

So LUB <=> IVT <=> Connectedness (Is this correct?)

3. Does the Complete Metric Space also have this property?

I am trying to find is a connection between Completion in Metric Space and Connectedness in Metric Space.

Also, Cauchy Completeness in R is weaker than LUB in R.(Cauchy Completion + Archimedian Property $implies$ LUB)

So does Completion in $mathbb{R}$ is stronger than Completion in Metric Spaces?

Basically I am trying to see the connections between all the the properties of Real Analysis and Metric Spaces.