My second question is, there is a main panel with a 100 amp double breaker and two 60 am double breakers. Prior owners run a line off the 60 amp double breaker to the wash room washer and dryer. Can I run a line with conduit 10 gauge line off of one of the 60 am double breakers? Do I just run a 10 gauge line off the 60 amp breaker to a new breaker box with a 30 amp double breaker? I appreciate all advice. Thank You.
The breaker MUST be sized to be equal to or less than what the wire permits in ADDITION to what the manufacture of the appliance says to use.*
The breaker can NOT be greater than what the wire permits. Period.
#10 gauge size wire is sized for at most 30 Amps. Therefore, at most a 30A breaker should be used.
Most washers require no more than a 20A circuit and new code requires them to be GFCI protected. I've never seen a double pole washing machine, but that doesn't mean they don't exist.
Your existing setup doesn't sound correct at all. But what I've outlined is pretty standard.
*The breaker/wire can be greater than what the manufacture calls for IF and only IF there is a fuseable disconnected/sub-panel to protect the appliance(s).
You can't hang 30A circuits off a 60A breaker. The only thing you can do with a 60A breaker - is run wire rated for 60A, to serve a device rated to draw 60A. One possible device is a sub-panel.
I gather from your other answer, that you're dealing with panels whose breakers are not available at any sane price or reliability, so changing breaker size is not an affordable option for you. You have a main 120/240V panel (this panel), and also two 120V-only panels each fed by a different pole of "hot".
If you can obtain a 30A breaker for that main panel, yes, you can connect a water heater in the usual way with 10 AWG wire. You don't need a neutral wire unless the water heater or timer is actually 120/240. If it's straight 240, 10-2 cable (black/white/ground) will suffice, relabel the white as a hot.
Otherwise: I would install a new sub-panel somewhere, perhaps in the washer-dryer area (cost of wire may be a factor). Feed the sub-panel off the 60A double breaker now used for the washer and dryer. Connect the sub-panel to the main panel with wire rated for 60 amps. If necessary, change wire - because for you, wire is cheaper than breakers.
Then, in the new sub-panel, install the correct breakers for washer (single 15-20A, GFCI) and water heater (typically double, 30A). If you also want to power other loads off this panel, ask here.
On choice of sub-panel: "What a fool I was to buy a panel with so many spaces" was said by no one, ever. Err on the side of "really big" when buying panels. With panels, it's OK if the panel is rated for more than 60 amps. The 60A breaker in the main panel will protect the wire to the panel. Some panels are sold in "combo packs" bundled with breakers, which is far-and-away the cheapest way to buy breakers+panel. If you have extra breakers, you can sell them on eBay or save them for the happy day when you replace those other panels. I'd buy this at a real electrical supply house, they will also have the best selection (and often better prices - electricians don't buy there for no reason).
On choice of wire: check your existing wire to see if it is legal in current code. I don't know what your local regulations require, what will easily install, and what's available at your local electrical supply. Normally I'd say consider aluminum wire in AA-8000 alloy; but given the age of your breakers I'd avoid it.
If you DIY, I would figure about $200 depending on the wire distance and the service panel you choose. If you can spend $30 extra and get more panel, definitely a good idea.