Mi aplicación tiene los siguientes parámetros:

android {
    compileSdkVersion 23
    buildToolsVersion "23.0.2"

    defaultConfig {
        applicationId "de.munich.myApp"
        minSdkVersion 17
        targetSdkVersion 23
        multiDexEnabled true
    }

    buildTypes {
        debug {
            debuggable true
        }
        release {
            minifyEnabled false
            shrinkResources true
            proguardFiles getDefaultProguardFile('proguard-android.txt'), 'proguard-rules.txt'
        }
    }
}

Incluí el paquete de soporte v7:

dependencies {
    compile 'com.google.code.gson:gson:2.4'
    compile 'com.android.support:appcompat-v7:23.2.0'
    compile 'com.google.android.gms:play-services:8.4.0'
    compile 'com.android.support:design:23.2.0'
    compile 'com.android.support:recyclerview-v7:23.2.0'
    compile 'com.android.support:cardview-v7:23.2.0'
}

Y ahora trato de iniciar mi aplicación en un Galaxy Ace con Android versión 4.4.2 (API nivel 19) y obtengo muchos errores porque no se encontraron algunos métodos a los que hace referencia android.support.v7:

Could not find method android.view.Window$Callback.onSearchRequested, referenced from method android.support.v7.internal.view.WindowCallbackWrapper.onSearchRequested
Could not find method android.view.Window$Callback.onWindowStartingActionMode, referenced from method android.support.v7.internal.view.WindowCallbackWrapper.onWindowStartingActionMode
Could not find method android.content.res.TypedArray.getChangingConfigurations, referenced from method android.support.v7.internal.widget.TintTypedArray.getChangingConfigurations
Could not find method android.content.res.TypedArray.getType, referenced from method android.support.v7.internal.widget.TintTypedArray.getType

Por supuesto, la aplicación se bloquea.

¿Hay alguna manera de solucionar esos problemas sin tener que aumentar elminSdkVersion?

EDITAR

import android.app.Notification;
import android.content.Context;
import android.support.v4.app.NotificationCompat;

public class HelperNotification {

    /**
     *
     * @param context
     * @param title
     * @param message
     * @return
     */
    public static NotificationCompat.Builder buildNotification(Context context, String title, String message) {
        NotificationCompat.Builder builder = new NotificationCompat.Builder(context);

        builder.setContentTitle(title);
        builder.setContentText(message);

        int defaults = 0;
        defaults = defaults | Notification.DEFAULT_LIGHTS;
        defaults = defaults | Notification.DEFAULT_VIBRATE;
        defaults = defaults | Notification.DEFAULT_SOUND;

        builder.setDefaults(defaults);

        return builder;
    }
}

EDITAR 2

I/dalvikvm: Could not find method android.os.UserManager.getApplicationRestrictions, referenced from method com.google.android.gms.common.zze.zzaq
W/dalvikvm: VFY: unable to resolve virtual method 1747: Landroid/os/UserManager;.getApplicationRestrictions (Ljava/lang/String;)Landroid/os/Bundle;
D/dalvikvm: VFY: replacing opcode 0x6e at 0x0012
E/dalvikvm: Could not find class 'android.app.AppOpsManager', referenced from method com.google.android.gms.common.zze.zzb
W/dalvikvm: VFY: unable to resolve check-cast 35 (Landroid/app/AppOpsManager;) in Lcom/google/android/gms/common/zze;
D/dalvikvm: VFY: replacing opcode 0x1f at 0x000e
I/dalvikvm: Could not find method android.content.pm.PackageManager.getPackageInstaller, referenced from method com.google.android.gms.common.zze.zzi
W/dalvikvm: VFY: unable to resolve virtual method 605: Landroid/content/pm/PackageManager;.getPackageInstaller ()Landroid/content/pm/PackageInstaller;
D/dalvikvm: VFY: replacing opcode 0x6e at 0x000b
I/GMPM: App measurement is starting up, version: 8487
I/GMPM: To enable debug logging run: adb shell setprop log.tag.GMPM VERBOSE
E/GMPM: GoogleService failed to initialize, status: 10, Missing an expected resource: 'R.string.google_app_id' for initializing Google services.  Possible causes are missing google-services.json or com.google.gms.google-services gradle plugin.
E/GMPM: Scheduler not set. Not logging error/warn.
E/GMPM: Uploading is not possible. App measurement disabled
D/dalvikvm: GC_CONCURRENT freed 265K, 16% free 8722K/10356K, paused 5ms+2ms, total 26ms
W/dalvikvm: VFY: unable to find class referenced in signature (Lde/hof/my_project/enums/status/images/StatusImageScale;)
W/dalvikvm: VFY: unable to resolve static field 34720 ($SwitchMap$de$hof$my_project$enums$status$images$StatusImageScale) in Lde/hof/my_project/fragments/FragmentSplash$1;
D/dalvikvm: VFY: replacing opcode 0x62 at 0x0000
E/dalvikvm: Could not find class 'de.hof.my_project.tasks.images.TaskImagesScale', referenced from method de.hof.my_project.fragments.FragmentSplash.onViewCreated
W/dalvikvm: VFY: unable to resolve new-instance 9154 (Lde/hof/my_project/tasks/images/TaskImagesScale;) in Lde/hof/my_project/fragments/FragmentSplash;
D/dalvikvm: VFY: replacing opcode 0x22 at 0x0003
D/dalvikvm: DexOpt: unable to opt direct call 0xf89d at 0x05 in Lde/hof/my_project/fragments/FragmentSplash;.onViewCreated

Agregué algunas salidas de registro mor. Esto es cuando ejecuto mi aplicación en un Samsung Galaxy S3 con API 17 (mi API mínima requerida). La semana pasada, antes de actualizar mis bibliotecas de 23.0 a 23.2, todo funcionaba. Ahora parece que no puede encontrar algunas de las clases que definí.