can I get a virus while my computer is in sleep mode?

If you mean in sleep mode, as in, while your computer is on standby, a bit like when it's switched off, then no you can't, that's the short answer anyway.

exceptions to that..

if a computer is directly connected to the internet, or you have a router which has many unpatched vulnerabilities, then it is possible for a magic packet to be sent to a computer, to power it on. a magic packet is legitimate method for poering your own computer on remotely, or say you are in a big building/organisation and you work in the IT dept and you need to remotely administer them, saves a hell of a lot of leg work, all that leg work can be replaced by a few seconds of clicking.

But of course, it can be used for bad purposes too.

In general the magic packet, or WOL (wake on LAN) technology is disabled on most computers. You switch it on or off via BIOS screen or device manager, (sometimes both), all hardware dependant, so you have to know what you are doing, or have done it a few times before.

Once a somputer is remotely powered on, the attacker (or robot/automated script/program) would then have to get pass your computers security, which is software dependant, so that is very complicated, unless you want to end up learning an awful (years worth) lot of technology, i would suggest not worrying about it, and concentrate on using a range of different but strong passwords for all the places you go, they don't need to be too different, just can't be exactly the same at the least. and make sure you have not opened any ports in your firewall, and no services are listening on any open ports. Make sure you backup your stuff regularly, and maybe consider getting a second laptop, to back your stuff onto, and every now and then, completely reinstall one of your computers to make sure any insecurities gained along the way are wiped.

Yes...because you are still conected to the internet and the screen saver is an executable program........

Not likely... you need to execute a file that is infected to get a virus.