I'm seeing the same issue and ran diag in the OSGi console.
I've found that the SonarLint eclipse bundles have references to 2.3.1 versions of the core SonarLint bundles, while the plug-in payload includes the 2.4.0 versions of the core packages.
For example: The diag command shows that the sonarlint.eclipse.core bundle is referencing sonarlint-client-api_2.3.1 while the plug-in payload includes sonarlint-client-api_2.4.0.
Note: I'm seeing the same version issue at the bottom of the diag window provided by the OP.
I'm thinking the SonarLint 2.2 bundle needs to be rebuilt/repackaged with the correct package versions.
Here's my OSGi console output for reference.
osgi> ss
...
1241 INSTALLED org.sonarlint.eclipse.core_2.2.0.201607291419-RELEASE
1242 INSTALLED org.sonarlint.eclipse.jdt_2.2.0.201607291419-RELEASE
1243 INSTALLED org.sonarlint.eclipse.ui_2.2.0.201607291419-RELEASE
1244 INSTALLED org.sonarsource.sonarlint.core.sonarlint-client-api_2.4.0
1245 INSTALLED org.sonarsource.sonarlint.core.sonarlint-core_2.4.0
osgi> diag 1241
reference:file:plugins/org.sonarlint.eclipse.core_2.2.0.201607291419-RELEASE.jar [1241]
Direct constraints which are unresolved:
Missing required bundle org.sonarsource.sonarlint.core.sonarlint-client-api_2.3.1.
Missing required bundle org.sonarsource.sonarlint.core.sonarlint-core_2.3.1.
Missing required capability Require-Capability: osgi.ee; filter="(&(osgi.ee=JavaSE)(version=1.8))".
Leaf constraints in the dependency chain which are unresolved:
reference:file:plugins/org.sonarsource.sonarlint.core.sonarlint-core_2.4.0.jar [1245]
Missing required capability Require-Capability: osgi.ee; filter="(&(osgi.ee=JavaSE)(version=1.8))".
reference:file:plugins/org.sonarsource.sonarlint.core.sonarlint-client-api_2.4.0.jar [1244]
Missing required capability Require-Capability: osgi.ee; filter="(&(osgi.ee=JavaSE)(version=1.8))".
osgi> diag 1242
reference:file:plugins/org.sonarlint.eclipse.jdt_2.2.0.201607291419-RELEASE.jar [1242]
Direct constraints which are unresolved:
Missing required bundle org.sonarlint.eclipse.core_0.0.0.
Missing required capability Require-Capability: osgi.ee; filter="(&(osgi.ee=JavaSE)(version=1.8))".
Leaf constraints in the dependency chain which are unresolved:
reference:file:plugins/org.sonarsource.sonarlint.core.sonarlint-core_2.4.0.jar [1245]
Missing required capability Require-Capability: osgi.ee; filter="(&(osgi.ee=JavaSE)(version=1.8))".
reference:file:plugins/org.sonarsource.sonarlint.core.sonarlint-client-api_2.4.0.jar [1244]
Missing required capability Require-Capability: osgi.ee; filter="(&(osgi.ee=JavaSE)(version=1.8))".
reference:file:plugins/org.sonarlint.eclipse.core_2.2.0.201607291419-RELEASE.jar [1241]
Missing required capability Require-Capability: osgi.ee; filter="(&(osgi.ee=JavaSE)(version=1.8))".
osgi> diag 1243
reference:file:plugins/org.sonarlint.eclipse.ui_2.2.0.201607291419-RELEASE.jar [1243]
Direct constraints which are unresolved:
Missing required bundle org.sonarlint.eclipse.core_0.0.0.
Missing imported package org.sonarsource.sonarlint.core.client.api.connected_0.0.0.
Missing imported package org.sonarsource.sonarlint.core.client.api.util_0.0.0.
Missing required capability Require-Capability: osgi.ee; filter="(&(osgi.ee=JavaSE)(version=1.8))".
Leaf constraints in the dependency chain which are unresolved:
reference:file:plugins/org.sonarsource.sonarlint.core.sonarlint-core_2.4.0.jar [1245]
Missing required capability Require-Capability: osgi.ee; filter="(&(osgi.ee=JavaSE)(version=1.8))".
reference:file:plugins/org.sonarsource.sonarlint.core.sonarlint-client-api_2.4.0.jar [1244]
Missing required capability Require-Capability: osgi.ee; filter="(&(osgi.ee=JavaSE)(version=1.8))".
reference:file:plugins/org.sonarlint.eclipse.core_2.2.0.201607291419-RELEASE.jar [1241]
Missing required capability Require-Capability: osgi.ee; filter="(&(osgi.ee=JavaSE)(version=1.8))".
osgi> diag 1244
reference:file:plugins/org.sonarsource.sonarlint.core.sonarlint-client-api_2.4.0.jar [1244]
Direct constraints which are unresolved:
Missing required capability Require-Capability: osgi.ee; filter="(&(osgi.ee=JavaSE)(version=1.8))".
osgi> diag 1245
reference:file:plugins/org.sonarsource.sonarlint.core.sonarlint-core_2.4.0.jar [1245]
Direct constraints which are unresolved:
Missing required bundle org.sonarsource.sonarlint.core.sonarlint-client-api_2.4.0.
Missing required capability Require-Capability: osgi.ee; filter="(&(osgi.ee=JavaSE)(version=1.8))".
Leaf constraints in the dependency chain which are unresolved:
reference:file:plugins/org.sonarsource.sonarlint.core.sonarlint-client-api_2.4.0.jar [1244]
Missing required capability Require-Capability: osgi.ee; filter="(&(osgi.ee=JavaSE)(version=1.8))".