Вопрос: Как перенести метод для кнопок в отдельный класс в WPF?

Лучший ответ: Изучай паттер MVVM и его используй.
Пример:
Код взял из боевого решения. Если будет непонятно, спрашивайтеpublic partial class SetColdBalance : Window
    {

        public SetColdBalance()
        {
            InitializeComponent();
        }
    }public class RelayCommand<T> : ICommand
    {
        #region Fields

        readonly Action<T> _execute = null;
        readonly Predicate<T> _canExecute = null;

        #endregion

        #region Constructors

        /// <summary>
        /// Initializes a new instance of <see cref="DelegateCommand{T}"/>.
        /// </summary>
        /// <param name="execute">Delegate to execute when Execute is called on the command.  This can be null to just hook up a CanExecute delegate.</param>
        /// <remarks><seealso cref="CanExecute"/> will always return true.</remarks>
        public RelayCommand(Action<T> execute)
            : this(execute, null)
        {
        }

        /// <summary>
        /// Creates a new command.
        /// </summary>
        /// <param name="execute">The execution logic.</param>
        /// <param name="canExecute">The execution status logic.</param>
        public RelayCommand(Action<T> execute, Predicate<T> canExecute)
        {
            if (execute == null)
                throw new ArgumentNullException("execute");

            _execute = execute;
            _canExecute = canExecute;
        }

        #endregion

        #region ICommand Members

        ///<summary>
        ///Defines the method that determines whether the command can execute in its current state.
        ///</summary>
        ///<param name="parameter">Data used by the command.  If the command does not require data to be passed, this object can be set to null.</param>
        ///<returns>
        ///true if this command can be executed; otherwise, false.
        ///</returns>
        public bool CanExecute(object parameter)
        {
            return _canExecute == null ? true : _canExecute((T)parameter);
        }

        ///<summary>
        ///Occurs when changes occur that affect whether or not the command should execute.
        ///</summary>
        public event EventHandler CanExecuteChanged
        {
            add { CommandManager.RequerySuggested += value; }
            remove { CommandManager.RequerySuggested -= value; }
        }

        ///<summary>
        ///Defines the method to be called when the command is invoked.
        ///</summary>
        ///<param name="parameter">Data used by the command. If the command does not require data to be passed, this object can be set to <see langword="null" />.</param>
        public void Execute(object parameter)
        {
            _execute((T)parameter);
        }

        #endregion
    }<Window x:Class="AdminTool.SetColdBalance"
        xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation"
        xmlns:x="http://schemas.microsoft.com/winfx/2006/xaml"
        xmlns:d="http://schemas.microsoft.com/expression/blend/2008"
        xmlns:mc="http://schemas.openxmlformats.org/markup-compatibility/2006"
        xmlns:i="http://schemas.microsoft.com/expression/2010/interactivity"
        xmlns:local="clr-namespace:AdminTool"
        mc:Ignorable="d"
        Title="Задать баланс"
        Height="150"
        Width="250"
        ResizeMode="NoResize"
        WindowStartupLocation="CenterScreen"">
    <Window.DataContext>
        <local:ColdWalletViewModel/>
    </Window.DataContext>
    <Grid>
     <Button Grid.Row="2" Width="100" Height="30" Content="Save" Command="{Binding SaveColdWallets}"/>
    </Grid>
</Window>public class ColdWalletViewModel : BindableBase
    {
        ICommand _saveColdWallet;
        public ICommand SaveColdWallets
        {
            get
            {
                return _saveColdWallet ?? (_saveColdWallet = new RelayCommand<object[]>((obj) =>
                {
                   ///Тут пишешь что должна выполнять твоя кнопка
                }), /*Тут можно написать условие при котором можно будет выполнить данную команду*/);
            }
        }
    }/// <summary>
    ///     Implementation of <see cref="INotifyPropertyChanged" /> to simplify models.
    /// </summary>
    public abstract class BindableBase : INotifyPropertyChanged
    {
        /// <summary>
        ///     Multicast event for property change notifications.
        /// </summary>
        public event PropertyChangedEventHandler PropertyChanged;

        /// <summary>
        ///     Checks if a property already matches a desired value.  Sets the property and
        ///     notifies listeners only when necessary.
        /// </summary>
        /// <typeparam name="T">Type of the property.</typeparam>
        /// <param name="storage">Reference to a property with both getter and setter.</param>
        /// <param name="value">Desired value for the property.</param>
        /// <param name="propertyName">
        ///     Name of the property used to notify listeners.  This
        ///     value is optional and can be provided automatically when invoked from compilers that
        ///     support CallerMemberName.
        /// </param>
        /// <returns>
        ///     True if the value was changed, false if the existing value matched the
        ///     desired value.
        /// </returns>
        protected bool SetProperty<T>(ref T storage, T value, [CallerMemberName] string propertyName = null)
        {
            if (Equals(storage, value))
            {
                return false;
            }

            storage = value;
            this.OnPropertyChanged(propertyName);
            return true;
        }

        /// <summary>
        ///     Notifies listeners that a property value has changed.
        /// </summary>
        /// <param name="propertyName">
        ///     Name of the property used to notify listeners.  This
        ///     value is optional and can be provided automatically when invoked from compilers
        ///     that support <see cref="CallerMemberNameAttribute" />.
        /// </param>
        protected void OnPropertyChanged([CallerMemberName] string propertyName = null)
        {
            this.PropertyChanged?.Invoke(this, new PropertyChangedEventArgs(propertyName));
        }
    }

Худший ответ: Вам не нужно переносить методы. Просто опишите их в нужном вам классе и вызовите в методах в классе окна.
Например так:public partial class MainWindow : Window
    {
        Animation animation;

        bool isOptionsOpen1 = false;
        bool isOptionsOpen2 = false;

        public MainWindow()
        {
            InitializeComponent();
            HotKeys hk = new HotKeys(this);
            animation = new Animation(this);
        }               

        private void OpenOptions(object sender, RoutedEventArgs e)
        {             
           animation.OpenOptions();
        }

        private void HideOptions(object sender, RoutedEventArgs e)
        {           
           animation.HideOptions();
        }
    }