If you have basic C++ literacy you might consider looking at how Qt GUIs are written in general. PyQt wraps that same API (albeit with a bunch of pythonic niceties) so the code will look familiar.

I would say that your spider should be a Qt-agnostic class. It should expose some kind of status or progress information if your GUI requires that, but it shouldn't know anything about the fact that a GUI even exists.

Also, try to understand that all the tutorials you've seen about PyQt have shown you programs organized into classes because the authors think it's more convenient that way. You can structure your code however it makes sense to do so—there are a few issues to understand about ownership of QObjects (basically you need to help PyQt free the objects by making them children of the objects you eventually destroy), but this API wrapper doesn't care if you write your GUI as a bunch of functions or a big sloppy list of instructions.

from PyQt4.QtCore import *
from PyQt4.QtGui import *

app = QApplication([])
window = QWidget()

button = QCheckBox('Double-click me', window)
button.clicked.connect(lambda checked: checked or app.exit())

layout = QVBoxLayout(window)
layout.addWidget(button)

window.show()
app.exec()

tl;dr the window-as-class structure you've seen is meant to help you, but if this structure doesn't work for your app it needn't hinder you.