As far as I know, no they cannot climb up vertical walls. They can run along basically anything horizontal or diagonal which is sturdy enough for them with ease.
It's very hard to mouse proof your home. You just need to walk around your entire house several times to make sure there are absolutely NO holes or cracks, as mice are very good at squeezing through tight spaces.
To catch a mouse you firstly need to know exactly where their main hideout is. Then getting a cardboard box put some food in it, tough enough and big enough so as they cannot drag it through the hole they create to get in, you may create the holes if you wish but the mice are able to just gnaw through the cardboard so it's not necessary. I suggest using a pizza box with a couple pieces of pizza still left in it as it does not crumble and is quite large so the mice cannot drag it out of the box. Place the box with food in it right in front of their hideout, wedge it in front of the hideout to keep it from going anywhere and to give the mice no other choice then to gnaw through the box. You can then come back to the hideout and cover the box in a blanket or something to avoid mice coming out of the box while you carry it away.
It's probably best to release the mice away from the suburban area and into the countryside near a bunch of leaves or a pile of wood. Leave some food scraps around the are you release them to support them for a while before they wonder off, probably to find another house. Try to make sure you release them not too close someone's block of land or house, to prevent them from finding somewhere else to get into.
Please try to avoid killing them as they too feel pain and should not have to be put through something so bad as being squished alive when they are only looking for a home and some food to be able to live.
Hopefully you will be able to catch and release the mice before having too many restless nights of worry that a mouse may run over your face. Try not to worry too much as it is highly unlikely for a mouse to approach a human in the first place, they usually run away from movement, even you breathing and moving in your bed during your sleep will make them avoid you as much as possible. Just be sure to keep food out of your bedroom and keep it very clean and tidy so that there is no need for them to go into your bedroom, where you sleep, in search for food and as you have wedged the box in front of their hideout they shouldn't even feel the need to leave the hideout.
Hope this helps and good luck!
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