B737 NG Captain reported a gate return for AUTO SLAT Fail. On takeoff; the Captain reported vertical guidance was lost along with unreliable air speed and the flight crew returned to land at departure airport.

Narrative

We were dispatched in an aircraft with multiple MEL items like the Captain's side PTT (Press To Transmit) Inop; LOGO lights Inop; and several other cabin items. After the initial push back from the gate; the tow truck was disconnected and during the after engines start checklist flow; the Six Pack was pushed to test to discover that the Overhead AUTO SLAT FAIL Amber Light was ON. QRH was complied with; and it showed that one of the dual channeled systems had failed. We contacted Maintenance Control via DISPATCH to notify them of our status. Maintenance Control recommended that we return to gate to have Maintenance Technicians trouble shoot which channel or system failed. We coordinated with Operations for a gate return. The AUTO SLAT FAIL failed system was deferred and MEL with the back up or secondary system operating normal. So; we pushed back the second time; started both engines; ran all the appropriate checklists; Operations check normal. Now on the runway; cleared for takeoff; thrust applied; TOGA button pushed; I as the Pilot Flying (PF) crossed check as always my PFD indicated airspeed with the Standby Indicator's airspeed as the First Officer who was the Monitoring Pilot (PM) called out 100. Air Speeds checked correct all around. However; right at rotation; I lost the vertical guidance of the Flight Director but I continued the rotation and takeoff as I called out vertical guidance lost. As I continued the climbout passing through 10;000 ft. and hand flying; I confirmed several times to making sure the airspeeds and aircraft attitude information matches that of the standby instrument and the FO's (First Officer). Confirming that the FO's flight director is intact and correct. I transferred control of the aircraft to the First Officer with his Flight Director as the MASTER and B autopilot switched ON as I resumed communications albeit my fumbling with an inop PTT and hand mike and running of the checklists. Just then I noticed the amber SPD LIM indication on my PFD. All the while the airspeed indicators were all reading the same. SPD LIM QRH could not be found but I knew the common denominator of what failed had to do with airspeed like minimum speed reversion. With the situation at hand; it did not occur to me to run the memory items for airspeed disagree because all the indicated airspeeds agreed. Not long after passing through FL200; we started to get intermittent short bursts of left stick shaker with minimum speed limit indicator popping out but the airspeed matches the rest. So; I took out the QRH and ran the Airspeed Disagree checklist.I decided in collaboration with the FO to return to ZZZ. Briefing was given to the Cabin Crew; Dispatch was notified; and priority handling was requested from ATC with a return to ZZZ. The FO did a very beautiful landing especially in an overweight status. We asked the fire truck to check our wheel wells including temperature checking our main wheels. The fire truck advised us that the temps are safe to continue taxi to the gate. For some reason I cannot recollect; I ran the Brakes Cooling Chart forgetting that it is for RTO only because I just wanted to make sure we are always in a safe operating mode.We taxied safely back to the gate with the fire trucks right behind us. Engines shut down; all checklists complied with; Dispatch; Maintenance Control and Operations were contacted.