Does battery condition have anything to do with macbook pro performance?
Yes, it could have something to do with the battery - and yes, it could be something else. In short, it's not possible to know without examining your actual hardware.

The "Replace Soon" warning combined with the fact that the battery lasts only for 1 hour really means that you should get the battery replaced. It is not going to get better by itself. Nor is unusual - getting 4.5 years of life out of a battery is actually quite good.

If the battery problem causes extra heat - either from the battery itself, charging circuits or elsewhere - it might be the source of your computer heating up.

The source of the heating up might also be something else entirely. For example it could be as simple as a broken or very dirty fan. Replacing the fan could fix the problem.