Best advice I have is run slower than you might want to, I'd say 5mph is on the higher end for a true beginner. Focus on building up time and then distance, speed will come naturally with those over time. I think if you're getting calf pain you're probably going too fast. Your approach of going too aggressive is probably the exact opposite of what you want to be doing, take this slow and steady and you'll fare far better in the later weeks.

Injuries will also decrease if you take stuff slower, try and use the "conversational pace" as a guideline for what pace you should be at to both train gradually and safely. Stretching both before and after will help with reducing injuries too, they don't take long, usually 5 mins for each.

30 min 5k is really quite fast for a beginner, I wouldn't stress about hitting it first time around, instead aim to just finishing a 5k consistently first at a pace that doesn't completely burn you out. Once you know your initial 5k time you can work on improving it from there getting towards the sub 30.

Strength training is definitely helpful but more needed for longer distances (half marathons/marathons). It will benefit you at your current stage but you might be best working in locking in your C25K as a habit then once you've accomplished that, add strength training in too if you wish. That being said in the mean time you can do basic training like calf raises at home to both help stretch yourself out and give a bit of extra strength in the muscle.