Looking at your second question first, if you look at the end of the fimplicit function's documentation page, there's a link to "ImplicitFunctionLine Properties". On that page, the value of the Type property is 'implicitfunctionline'. By the "Specify Default Values" section on this documentation page that means you should set the property:
P = "default" + "implicitfunctionline" + "LineWidth"
This sort of implicitly answers your first question. Let's try it.
h = fimplicit(@(x,y) x.^2 - y.^2 - 1);
d = h.LineWidth % show the factory default
title("LineWidth of " + d)
set(groot, P, 10*d);
figure
h = fimplicit(@(x,y) x.^2 - y.^2 - 1);
d = h.LineWidth % show the factory default
title("LineWidth of " + d + " after changing default")
For your third question I believe the answer is no. However, if you're okay with changing the properties of all objects with a LineWidth property after the fact, you could use the findall or findobj functions and ask for all objects that have a LineWidth.
h = findall(groot, ... % All graphics objects that are "descendants" of groot (i.e. all of them)
'-property', 'LineWidth') % that possess a LineWidth property
Note that in addition to the lines created by fimplicit h lists two axes (which the axes properties page states controls "Line width of axes outline, tick marks, and grid lines, specified as a positive numeric value in point units. One point equals 1/72 inch.") and two text objects (the titles, for which the text properties page says the LineWidth property controls "Width of box outline, specified as a scalar numeric value in point units. One point equals 1/72 inch.")
So let's look at what would happen if you changed all those LineWidths to something much larger. It might not look exactly as you want, since the axes has a LineWidth.
f = figure;
h = fimplicit(@(x,y) x.^2 - y.^2 - 1);
t = title('Example');
titleLineWidthBefore = t.LineWidth
hasLineWidth = findall(f, '-property', 'LineWidth')
set(hasLineWidth, 'LineWidth', 10)
The LineWidth change did affect the text object returned by title, but since that object's EdgeColor is 'none' we don't see any visible change.
t.EdgeColor
titleLineWidthAfter = t.LineWidth
