https://www.tubesandmore.com/sites/default/files/schematics/crosley_corp_j30bc_pg12-1.pngFirst thing is replace the 10 mfd / 15 mfd/ 15mfd electrolytics shown between the 5y3 and the speaker connections on the schematic.
These will be rated at 450volts . After that all the ones such as .1mfd , .02 etc that are paper and those need to come out and be replaced with modern same or near the capacity marked on the old ones. Complete each one before moving on ( Important!). At that point after making sure nothing is touching each other or shorted from handling, you can usually apply AC , stand back, the majority of old radios will work with this minimum. Once you get it working and picking up a station , other issues like intermittents or poor contacts can be worked out. There may be out of spec resistors in a Radio that old here and there but not so bad as to keep it from working and can be dealt with later. When replacing the caps cut the leads near the body and twist / solder to the old. This isn't the prettiest way , but i recommend don't disturb the larger joints with multiple connections . No "cold" solders , heat the work first . Observe polarity on the electrolytics , the arrow points to negative rail. If you put these in backwards you'll get a nasty surprise. Don't lose focus , do this when there's no chance you'll be interrupted . Overhandling (sp?) can be A problem , you'll see what I mean. Go slow. Be careful when there's power , the trans is probably 250v and definitely noticeable.
A pair of medical forcepts will be your best friend.
Don't try to run it with any of the 80 year old caps, if they short ( and they will, not debatable) it could ruin things like expensive transformers and whatnot .
The speaker should be fine unless it's obviously damaged or punched out. You need to use the original due to its field coil rather than a permanent magnet
This is basically the non technical approach, there's a phone book worth of retro science in these things but it's not really necessary to know , you're just replacing components.
http://www.tuberadios.com/capacitors/I usually get the Radio working first before I do any cosmetics. For the AC cord replacement I just buy an extension cord and cut off the end , or if you want to use that antique plug , it probably just uses screws to attach. The grill cloths are available , maybe not an exact match but no one will know . It's probably glued and sandwiched like you said, to keep it tight. You'll have to figure out how to get it off . The one thing i always say to myself about these things is- if it went in , it will come out.
http://www.radiodaze.com/grille-cloth/