You could save your Shapefile as a Spatialite file, see end of this answer.
However to answer the question you have asked about isolating polygons, and if working with your data as a shapefile, then really the first thing to do, as Kazuhito commented above, is to create a spatial index on your shapefile layer.
Double click on the layer, or right-click > properties, then in the
General tab. Click on
Create spatial index
This will dramatically increase the performance of QGIS handling your shapefile layer.
Again, building upon one of the comments, if you know the polygons you want to edit you can
Filter the shapefile to only show that polygon
Right-click the layer and select
Filter to open up a query builder
Double click on the field you want to filter by to add it to the query window, or type it, and build up an expression as in the screenshot.
If you don't know any attributes of the polygon the zoom in and use the `Identify feature'
Another way to isolate a few polygons, as is the question, and again if you know some attributes of the polygons you want, is to
Select by expression and the
Zoom to selection.
Again build an expression to select the polygons you want
Then 'Zoom to selection'
You could save your Shapefile as a Spatialite file
Layer > Save as and select
Spatialite in the drop down format. Browse to where to save it to, and name it. Make sure
Add saved file to map is selected.
Spatialite is a database format, and so editing will be inherently quicker. With so many polygons it's not a bad idea anyway as it will give you database capabilities on your data. If you need to convert back to a shapefile you can do this by reversing the process and save as a shapefile.
After your recent edit I now think I better understand. I will add it to this answer for now, and then if this works I shall tidy up the answer so that it better reflects your edited question.
One method, which I don't think is actually what you want but I shall mention it first, is that you can use Vector > Geometry Tools > Multipart to singlepart, but this will separate all of them into separate individual polygons, so I suspect that this is not what you want?
To edit just two parts, as in your image...
In edit mode, with no polygons selected, use the split feature tool to cut each polygon part that you want to work with. This will make them separate polygons.
You can then edit those parts separately so that the nodes of each edge you want to join are over each other. You can snap the nodes over each other, by switching on snapping.
If the polygons are touching like this then you can select only those two polygons that you want to join using Vector > Geoprocessing Tools > Dissolve.
Browse to where you want to save the output as a temporary shapefile, and OK.
This will 'dissolve' the two polygons together into a separate layer. You can delete the original two polygons from your original layer, and then copy and paste the dissolved polygon into the original layer.