Looks like you need to set your ‘Stroke Drawing Order’ to ‘Column Strokes in front’. You can do this under the ‘Table Setup’ options. The option for the ‘Stroke Drawing Order’ is located at the bottom of that window.

As for selecting a table to access its options. There are a few ways:

One way is to mouse over the left edge or top edge of your table until you see the table arrow indicator, then click, now you should have a row or column selected, then simply press SHIFT+ALT+CMD+B (on a Mac) to bring up the table options (or right click on the selected area and choose ‘Table Setup > Table Options’.

Another way is to select just a cell by clicking in a cell so that you can see the flashing cursor, then press ESC, then press SHIFT+ALT+CMD+B to bring up the table options.

TIP: If you want to apply certain formatting to the whole table without using the Table Options dialog window, you can select the entire table by mousing over the top left corner of your table and when you see a -45º arrow you can select the entire table. You can also select the entire table by using the shortcut ALT+CMD+A, but you must have at least one cell selected first.

You could try adding the InDesign logos to a CC library and then placing in Illustrator from the CC library. Then embed the logos in Illustrator to work with them.