If ATC doesn't assign an obstacle departure procedure in an IFR clearance, what should you do?
To answer your question directly, yes you can use the ODP to depart an airfield regardless of if ATC has specifically given it to you. You may use it until you reach the prescribed altitude in your IFR clearance or the beginning of the cleared routing. However, if ATC has given you specific instructions for your departure, you have to either accept those instructions and follow them. Or, decline the clearance and ask for what you need. After all, you are the Pilot in Command solely responsible for the safety of your own flight.
That being said, I have never heard an IFR clearance where the departure procedure is not explicitly spelled out. It does not have to be in the form of an official charted Obstacle Departure Procedure or complex instructions. As a matter of fact, it is usually a fairly simple instruction in order to put you in a position for ATC Departure to pick you up on Radar. It should sound like this:

C - Cleared to [your destination]
R - Fly runway heading. Radar vectors to the [blank] departure. Then as filed.
A - Altitude 3000 [or so]. Expect 10,000 in 10 minutes.
F - [Frequency you will use for departure].
T - Squawk [transponder code].

The “fly runway heading” statement means that you will fly the course of the extended center line of the runway of departure. And, continue to do so until told otherwise by ATC.
They may replace that statement with “Upon departure, turn right [blank] heading. This generally means you have the leeway to begin your turn once you are beyond the traffic pattern and/or above 1000 feet AGL.
If they stipulate for you to make an immediate turn to a specific heading as practical, do so as soon as you are above a safe altitude to bank the aircraft. I would generally make the turn when I have reached the Departure End of the Runway and above 400 feet AGL. ATC may give you an altitude at which to make that turn. In which case, make the turn at that altitude regardless of if you have reached the DER or not. Typically, ATC will not have you make the turn lower than 400 feet AGL. They would deem this as the practicable altitude for a turn. In an emergency or urgent situation, ATC may request a turn as low as 200 feet AGL. They would not want you banking low and slow in IMC.
If ATC does not give you a specific set of departure procedure instructions, the FAA recommends in the AIM that Part 91 flights should follow the charted ODP if one is available. This is only a recommendation and not a requirement. I would highly recommend informing ATC if you plan to amend their clearance instructions by departing on the ODP.
Something you may want to consider is that the numbers you alluded to in your questions are bare minimum performance values for certification of a Departure Procedure. You, as the PIC, should not be following these numbers. If your aircraft can not achieve 35 feet AGL way before the DER, and a climb rate of 200 feet per Nautical Mile, you probably should not be flying that aircraft, at that airfield, on that day, in those conditions (weight, weather, etc).
Below are the FAA recommendations for this found in the Aeronautical Information Manual Chapter 5-2-9:

a. Instrument departure procedures are preplanned instrument flight rule (IFR) procedures which provide obstruction clearance from the terminal area to the appropriate en route structure. There are two types of DPs, Obstacle Departure Procedures (ODP), printed either textually or graphically, and Standard Instrument Departures (SID), always printed graphically. All DPs, either textual or graphic may be designed using either conventional or RNAV criteria. RNAV procedures will have RNAV printed in the title; for example, SHEAD TWO DEPARTURE (RNAV). ODPs provide obstruction clearance via the least onerous route from the terminal area to the appropriate en route structure. ODPs are recommended for obstruction clearance and may be flown without ATC clearance unless an alternate departure procedure (SID or radar vector) has been specifically assigned by ATC. Graphic ODPs will have (OBSTACLE) printed in the procedure title; for example, GEYSR THREE DEPARTURE (OBSTACLE), or, CROWN ONE DEPARTURE (RNAV) (OBSTACLE). Standard Instrument Departures are air traffic control (ATC) procedures printed for pilot/controller use in graphic form to provide obstruction clearance and a transition from the terminal area to the appropriate en route structure. SIDs are primarily designed for system enhancement and to reduce pilot/controller workload. ATC clearance must be received prior to flying a SID. All DPs provide the pilot with a way to depart the airport and transition to the en route structure safely.
e. What criteria is used to provide obstruction clearance during departure?

Unless specified otherwise, required obstacle clearance for all departures, including diverse, is based on the pilot crossing the departure end of the runway at least 35 feet above the departure end of runway elevation, climbing to 400 feet above the departure end of runway elevation before making the initial turn, and maintaining a minimum climb gradient of 200 feet per nautical mile (FPNM), unless required to level off by a crossing restriction, until the minimum IFR altitude. A greater climb gradient may be specified in the DP to clear obstacles or to achieve an ATC crossing restriction. If an initial turn higher than 400 feet above the departure end of runway elevation is specified in the DP, the turn should be commenced at the higher altitude. If a turn is specified at a fix, the turn must be made at that fix. Fixes may have minimum and/or maximum crossing altitudes that must be adhered to prior to passing the fix. In rare instances, obstacles that exist on the extended runway centerline may make an “early turn” more desirable than proceeding straight ahead. In these cases, the published departure instructions will include the language “turn left(right) as soon as practicable.” These departures will also include a ceiling and visibility minimum of at least 300 and 1. Pilots encountering one of these DPs should preplan the climb out to gain altitude and begin the turn as quickly as possible within the bounds of safe operating practices and operating limitations. This type of departure procedure is being phased out.

Note: “Practical” or “feasible” may exist in some existing departure text instead of “practicable.”