I just started out mining, and I'm using Nicehash to mine BTC, so my question is if I ran the Nicehash client on two different computers in my house, both using the same wallet address, does it add the profit to the wallet? What are the problems with it if i do use this?
You could use the same address with both miners and there would be no problem. Addresses can be reused, even though this is not recommended due to privacy concerns.
However, you will never manage to mine any bitcoins with home computers. They're too slow and don't contribute enough proof of work to get paid.
Yes, NicehHash allows you to mine whatever algorithm you like, and be paid out in Bitcoin, in fact I'd recommend only using one address for as many miners/computers as you have, they currently only pay out on balances over 0.1 BTC daily, and 0.01 BTC weekly (0.001 for their own wallets). So hitting these limits on multiple accounts may be difficult without losing some dust.
As @Murch says, you really shouldn't be reusing addresses, but in cases like this, it's difficult not to, since it's tricky to stop mining when you hit the payout limit (although not impossible, you just have to know when it is) and switch to a new payout address. If you're just getting started, then you can ensure that you don't use this address for other things, solely for NiceHash mining.
Some links on NiceHash payouts:
(note: these are links to "new" as they have recently updated their website so may break.)