Two things are at play here, when your IFR clearance takes effect and when the FAA assumes responsibility for your terrain and obstruction clearance.
If your clearance is "Cleared to XYZ as filed, climb and maintain seven thousand" without any further qualifiers, then you are cleared IFR. Full stop, end of story, that is an IFR clearance. In order to depart into Class E you also need to obtain an IFR release, but assuming you have both the clearance and the release you are perfectly legal to go blasting off from the airport, even in IMC, in fact even in 0/0 conditions (at least under Part 91 rules).
ATC is allowed to issue IFR clearances that pass through uncontrolled airspace (see, e.g., JO 7110.65 4–3–2b1, "Specify the destination airport [as the clearance limit] when practicable, even though it is outside controlled airspace.") Once you have your IFR release ATC will restrict IFR departures out of and arrivals in to your airport—although it is possible another aircraft may be operating IFR without a clearance, unless there is a Class E surface area. ATC cannot and will not guarantee IFR traffic separation in uncontrolled airspace, even if your clearance takes you through such airspace.
To sum it up: Provided you are properly certificated and current and everything else, you are permitted to fly IFR and also to enter IMC in Class G airspace immediately upon departure, whether or not you have received an IFR clearance. In order to fly IFR in Class E airspace (whether IMC or VMC) you need to receive a clearance, which you have in fact received. Everything about your scenario is perfectly legal as far as that goes.
As for terrain and obstruction clearance, if and only if your airport has an official Instrument Approach Procedure, then you can look it up in the Terminal Procedures booklet and see what (Obstacle) Departure Procedures it has. Each runway will have an (O)DP listed, or it will be listed as "N/A" (not authorized) for some reason. In the absence of either one, the standard departure procedure applies: cross the departure end of the runway at least 35' AGL, then continue to 400' AGL before starting a turn, and then continue climb at a rate of 200' per nautical mile or greater up to the minimum IFR altitude.
If your airport does not have any official IAP, then nobody has surveyed it to see what, if any, (O)DP is necessary. Instead you are the final authority as to whether a departure into IMC is safe, and what sort of procedure you would need to fly in order to ensure obstacle clearance. See this question already linked by quietflyer and my answer to it for further discussion.
ATC's rules do not distinguish between a Class G airspace that has been surveyed for instrument departures and one that has not. The phraseology should be the same in either case. It is entirely the pilot's responsibility to know the difference and how it affects your ability to safely depart into IMC.
Once they grant your IFR release ATC will be "blocking" the airspace in the immediate vicinity of the airport up to the initial altitude in your clearance. This means they are protecting that whole area so when you pop up they don't have an immediate loss of separation with another IFR aircraft nearby. This also means you have a little bit of wiggle room to maneuver before turning directly East in accordance with your clearance, but telling them what your plan is would probably be appreciated.