1. 即使我常常看医生，我也会生病。

2. 即使你不迟到，我也不想等你。

哈哈好傲娇啊233

既...又／也／还

1. 他们既一起去看电影，也一起喝咖啡

语法上没什么大问题但总感觉哪里怪怪的

2. 她们俩既喜欢谈论她们的同学，还羡慕一个男生。

Любят говорить о товарищах и завидуют молодого человека是这个意思吗？Apologies for my poor Russian...

即使。。。也 means  even if .
In sentence 1, 即使 is usually used to express the condition while 也 is often used for an unexpected consequence. I believe you did quite well here. But i have to say that there is something wrong with the phrase 可以, which is mainly used for permission  instead of possibility.
 For example, 你可以走了 ，in English, you may leave now .
So it can not be used together with 生病。Here a  better alternative of 可以should be 会, which is used to express possibility。Sometime, 会 is often used together with a negative word 不。Other examples:
即使所有人都离开你，我也会一直陪伴着你。（I will stay here with you even if all the other people leave you）
即使你来了，门也打不开。（The door cannot be opened even if you come here）
即使我病了，我也不会去看医生。(I won't go to the doctor even if I am ill.
即使你邀请（invite）我，我也不会去参加你的生日晚会。(I will not go to your birthday party even if yo invite me)