can my landlord charge me rent if my partner is hospitalized, meaning do they need a proof letter from the hospital?

You live there, you have to pay rent, period. The ENTIRE rent needs to get paid. Your medical issues do not change that.

Rent is to be paid, period. Bills and rent don't stop or decrease because someone responsible to pay them is hospitalized.

"I am only aloud to pay my own side.."  What?  If you're on the lease then you are jointly and individually responsible for the rent.  If your mother can't pay then you have to pay it all.

your mother must pay rent whether she is in a hospital or not

Assuming you are both on the lease, then you are both fully responsible for the rent.  Your likely options are to pay the whole thing (get what you can from your mother), or eventually get evicted.  

If the lease is up then you could exercise a 3rd option and give notice and move, but you will still have to pay all rent (somehow) up until you are able to move out. 

If there is no way you can afford the rent you could try requesting the landlord to let you out of the lease. Maybe they'll do it if you found a replacement tenant or if you paid a fee (1-2 months rent) but they might just refuse to let you out period.

First of all it is allowed - not aloud.

The LL doesn't care who does or doesn't pay the rent as long as it is paid.  If not paid in full when due, expect to be evicted.  Leasing is a business; the LL isn't your parent.  You or your mother signed a lease which is a legally binding document.  It covers the amount of rent and when that amount is due.

If the rent isn't paid, you'll be evicted.

If you are both on the same lease then there is no such thing as your part or her part. You are both equally responsible for the rent in full.  Her going to the hospital legally has nothing to do with the rent. It still must be paid in full & on time. 

If you are on separate leases then they cannot make you pay her rent at all.  You are only responsible for your rent stated on your lease. 

If you are a minor (a child), then no.  Your mother owes the rent.  If you are adult, then you owe the rent.  If your mother won't pay, kick her out.  However, that's your business whether you want to or not but if she doesn't pay, then you have to, or you will both be evicted.  How you and your mother decide to share the rent is your business, not the landlord's.  His only concern is that he receives ALL of it.  What portion each of you pays is up to you and your mother.

if you are both on the lease both are responsible. if only half is paid both are late. not 50/50..