This reads like one of those heartbreaking cases where the two people grow up at different speeds / in different directions compared to when they were 14 or 16.
You have matured and taken on increasing levels of responsibility as the years went by. You’re conscientious in your studies, apply yourself at work, and still want to help those around you by trying to alleviate their emotional load. This speaks to your maturity.
He, sadly, hasn’t grown up just yet. I have seen this happen many, many times with my couple friends who got together really young. Sometimes it works, but many time the 2 people just don’t progress at the same rate. He has not developed into the conscientious, hard-working person you have become. His current maturity level does not match yours.
OP, I know you love him and want to help him, but taking on his share of the housework, writing his emails, organising support for him, is NOT your job. And it’s not something that will ever, ever help him in the long run.
It’s sad because when these issues arise, it doesn’t mean there is no love or commitment anymore. I have learnt to understand that sometimes it’s just inevitable that people will grow and take responsibility at different speeds. He might need a kick up his arse which will make him get on with it (doesn’t sound like it, since you’ve been trying to do this for months). Or he might just be one of those people who instead of maturing at 21/22 when life starts to get serious, will mature at 29/30. And that’s OK.
Please make sure you look after yourself in these last few months of your degree too. Use the summer months, or whenever your placement gives you a break, to think long and hard whether he would really want to commit to an adult, responsible life in your marriage. Paying bills, taking on debts / mortgages, making long-term plans and having children - are these things he’ll responsibly and maturely react to?
The lying is bad. Don’t let yourself become more and more entangled in this man’s life if he doesn’t show any initiative to stop the lying and own his actions. You are too young, and you sound too great!